# Multiple collisions of eigenvalues and singular values of matrix Gaussian
field

Wangjun Yuan Department of Mathematics, University of Luxembourg. E-mail:
<EMAIL_ADDRESS>
###### Abstract

Let $X^{\beta}$ be a real symmetric or complex Hermitian matrix whose entries
are independent Gaussian random fields. We provide the sufficient and
necessary conditions such that multiple collisions of eigenvalue processes of
$A^{\beta}+T_{\beta}X^{\beta}T_{\beta}^{*}$ occur with positive probability.
In addition, for a real or complex rectangular matrix $W^{\beta}$ with
independent Gaussian random field entries, we obtain the sufficient and
necessary conditions under which the probability of multiple collisions of
non-trivial singular value processes of
$B^{\beta}+T_{\beta}W^{\beta}\tilde{T}_{\beta}$ is positive. In both cases,
the size of the set of collision times is characterized via Hausdorff
dimension.

AMS 2020 subject classifications: 15B52, 60B20, 60G15, 60G60, 15A18

Keywords and phrases: random matrix; Gaussian random field; Hitting
probabilities; singular values; eigenvalues.

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 Assumptions
    3. 1.3 Main results
    4. 1.4 Notations
  2. 2 Preliminaries
    1. 2.1 Hausdorff dimension and capacity
    2. 2.2 Hitting probabilities
    3. 2.3 Manifolds and Lie group
  3. 3 Multiple collisions of eigenvalue processes
    1. 3.1 The real case: proof of Theorem 1.1
    2. 3.2 The complex case: proof of Theorem 1.2
  4. 4 Multiple collisions of singular value processes
    1. 4.1 The real case: proof of Theorem 1.3
    2. 4.2 The complex case: proof of Theorem 1.4

##  1 Introduction

###  1.1 Background

The study eigenvalue processes of self-adjoint matrix-valued process can be
dated back to at least [Dyson1962]. Let $B(t)$ be an $d\times d$ real or
complex matrix Brownian motion, i.e. $B(t)$ is an $d\times d$ matrix whose
entries are i.i.d. real or complex standard Brownian motions. We define the
self-adjoint matrix Brownian motion by

$\displaystyle
X^{\beta}(t)=\dfrac{1}{\sqrt{2\beta}}\left(B(t)+B(t)^{*}\right)+X^{\beta}(0),$
(1.1)

where $\beta=1$ corresponds to the real case and $\beta=2$ is for the complex
case. It was obtained in [Dyson1962] that the set of (ordered) eigenvalue
processes $\\{\lambda_{i}^{\beta}(t):1\leq i\leq d\\}$ of $X^{\beta}(t)$
solves the following stochastic differential equation (SDE):

$\displaystyle
d\lambda_{i}^{\beta}(t)=\dfrac{\sqrt{2}}{\sqrt{\beta}}dW_{i}(t)+\sum_{j:j\neq
i}\dfrac{1}{\lambda_{i}^{\beta}(t)-\lambda_{j}^{\beta}(t)}dt,\quad 1\leq i\leq
d,t\geq 0,$ (1.2)

where the family $\\{W_{i}(t):1\leq i\leq d\\}$ is a collection of real-valued
i.i.d. standard Brownian motions. The solution to SDE (1.2) is known as Dyson
Brownian motion. Note that the SDE (1.2) is singular whenever collision of
eigenvalue processes happens. Hence, the non-collision property among the
eigenvalue processes is required to establish this theory. The key is to
obtain the boundedness of the following quantity

$\displaystyle\mathbb{E}\left[-\sum_{i\neq
j}\ln\left|\lambda_{i}(t)-\lambda_{j}(t)\right|\right]$ (1.3)

using Itô calculus and martingate theory. This argument is sometimes known as
McKean’s argument [McKean1969]. We refer to [anderson2010, Yau2017] for more
details. We remark that the solution to SDE (1.2) can be viewed as a particle
system, and is known as Coulomb gas model. See [Mehta2004, Forrester2010].

In the area of statistics, the singular value processes of matrix-valued
Gaussian processes can be used to study the stability in principal component
analysis. Let $d_{1}\leq d_{2}$ be two integers and let $B(t)$ be a
$d_{1}\times d_{2}$ matrix whose entries are independent real standard
Brownian motions. Consider

$\displaystyle W(t)=W(0)+B(t).$ (1.4)

and denote by $\\{s_{i}(t):1\leq i\leq d_{1}\\}$ the set of ordered singular
value processes of $W(t)$. If we write $\lambda_{i}(t)=s_{i}^{2}(t)$, then the
set $\\{\lambda_{i}(t):1\leq i\leq d_{1}\\}$ is the ordered eigenvalue
processes of $W(t)(W(t))^{*}$, which is known as Wishart process. It is shown
in [Bru1989] that the set of eigenvalue processes $\\{\lambda_{i}(t):1\leq
i\leq d_{1}\\}$ solves the following SDE:

$\displaystyle
d\lambda_{i}(t)=2\sqrt{\lambda_{i}(t)}dW_{i}(t)+ndt+\sum_{j:j\neq
i}\dfrac{\lambda_{i}(t)+\lambda_{j}(t)}{\lambda_{i}(t)-\lambda_{j}(t)}dt,1\leq
i\leq d,t\geq 0,$ (1.5)

where the family $\\{W_{i}(t):1\leq i\leq d\\}$ is a collection of i.i.d. real
standard Brownian motions. Similar to the eigenavlue processes of symmetric
matrix Brownian motion case, the SDE (1.5) is singular when eigenvalue
processes collide. The non-collision of eigenvalue processes was established
in [Bru1989] via the McKean’s argument. The solution of the SDE (1.5) is
sometimes called the squared Bessel particle system. We refer the readers to
[Graczyk2019] for more studies on the squared Bessel particle system.

Since last decade, there are a lot of works on the eigenvalue processes of
self-adjoint matrix-valued Gaussian processes and singular values of matrix-
valued Gaussian processes. Let $\\{B_{ij}^{H}(t):1\leq i\leq j\leq d\\}$ be a
family of standard fractional Brownian motions with Hurst parameter
$H\in(1/2,1)$. The $d\times d$ symmetric matrix $X^{H}(t)$, whose entries are
given by

$\displaystyle X^{H}_{i,j}(t)=\begin{cases}B_{ij}^{H}(t)+X_{ij}(0),&1\leq
i<j\leq d,\\\ \sqrt{2}B_{ii}^{H}(t)+X_{ii}(0),&1\leq i=j\leq d,\end{cases}$
(1.6)

is called the symmetric matrix fractional Brownian motion. It was obtained in
[Nualart2014] that the set of eigenvalue processes
$\\{\lambda_{i}^{H}(t):1\leq i\leq d\\}$ satisfies the following SDE:

$\displaystyle d\lambda_{i}^{H}(t)=dY^{H}_{i}(t)+2H\sum_{j:j\neq
i}\dfrac{t^{2H-1}}{\lambda_{i}^{H}(t)-\lambda_{j}^{H}(t)}dt,\quad 1\leq i\leq
d,t\geq 0,$ (1.7)

where for each $i$, $Y^{H}_{i}(t)$ is the sum of Skorohod integrals with
respect to all $B_{kh}^{H}(t)$. The eigenvalue SDE (1.7) also has singularity
when eigenvalue processes collide. By considering the quantity (1.3) together
with fractional calculus, [Nualart2014] established the non-collision of the
eigenvalue processes.

Let $B^{H}(t)$ be a $d_{1}\times d_{2}$ matrix whose entries are i.i.d. real
standard fractional Brownian motion with Hurst parameter $H\in(1/2,1)$, and
let $W^{H}(t)=W(0)+B^{H}(t)$. Then the matrix process $W^{H}(t)(W^{H}(t))^{*}$
is known as the fractional Wishart process. In [Pardo2017], the set
$\\{\lambda_{i}^{H}(t):1\leq i\leq d\\}$ of its eigenvalue processes was shown
to solve the following singular SDE:

$\displaystyle d\lambda_{i}^{H}(t)=dY^{H}_{i}(t)+2Hd_{2}dt+2H\sum_{j:j\neq
i}\dfrac{\lambda_{i}^{H}(t)+\lambda_{j}^{H}(t)}{\lambda_{i}^{H}(t)-\lambda_{j}^{H}(t)}t^{2H-1}dt,\quad
1\leq i\leq d,t\geq 0,$ (1.8)

where for each $i$, $Y^{H}_{i}(t)$ is the sum of Skorohod integrals. The non-
collision of eigenvalue processes of fractional Wishart process was
established in [Pardo2017] by a similar startegy as [Nualart2014].

Very recently, the eigenvalue processes of symmetric matrix Brownian sheet
were studied in [Yuan2023]. Let $\\{B_{ij}(s,t):1\leq i\leq j\leq d\\}$ be a
family of standard Brownian sheets and let $X(s,t)$ be the $d\times d$
symmetric matrix, whose entries are given by

$\displaystyle X_{i,j}(s,t)=\begin{cases}B_{ij}(s,t)+X_{ij}(0),&1\leq i<j\leq
d,\\\ \sqrt{2}B_{ii}(s,t)+X_{ii}(0,0),&1\leq i=j\leq d.\end{cases}$ (1.9)

Then $X(s,t)$ is known as the symmetric matrix Brownian sheet. With the help
of stochastic calculus on the plane, the stochastic partially differential
equation (SPDE) satisfied by the eigenvalue processes of $X(s,t)$ was found in
[Yuan2023]. The SPDE is again singular when collision of eigenvalue processes
happens. Unfortunately, as a direct consequence of [Jaramillo2018], the
eigenvalue processes of the symmetric matrix Brownian sheet $X(s,t)$ collide
with positive probability, which means that the SPDE is only able to describe
the eigenvalue processes until the first collision time.

From the models mentioned above, it is a common phenomenon that the stochastic
equations for the eigenvalue processes (singular value processes, resp.) of
self-adjoint matrix Gaussian processes (matrix Gaussian processes, resp.) are
singular when eigenvalue processes (singular value processes, resp.) collide.
So the equations are only able to describe the eigenvalues processes or
singular value processes until the first collision time. Thus, it is important
to know whether the eigenvalue processes or singular value processes collide
or not.

In addition to the McKean’s argument, a new method for eigenvalue processes
appears recently in [Jaramillo2018], which is an application of the hitting
probability of Gaussian random field.

Let $N\in\mathbb{N}$ be fixed and consider a real-valued centered continuous
Gaussian random field $\xi=\\{\xi(t):t\in\mathbb{R}_{+}^{N}\\}$ defined on a
probability space $(\Omega,\mathcal{F},\mathbb{P})$ with covariance given by

$\displaystyle\mathbb{E}\left[\xi(s)\xi(t)\right]=C(s,t),$

for some non-negative definite function
$C:\mathbb{R}_{+}^{N}\times\mathbb{R}_{+}^{N}\rightarrow\mathbb{R}$. For
$a=(a_{1},\ldots,a_{N}),\,b=(b_{1},\ldots,b_{N})\in\mathbb{R}_{+}^{N}$
satisfying $a_{i}\leq b_{i}$ for $1\leq i\leq N$, let $I=[a,b]$ be the compact
interval defined by

$\displaystyle
I=[a,b]=\prod_{j=1}^{N}[a_{j},b_{j}]\subseteq\mathbb{R}_{+}^{N}.$ (1.10)

Throughout this paper, we denote by $\iota=\sqrt{-1}$ the imaginary unit. For
$\beta\in\\{1,2\\}$, and $d\in\mathbb{N}$ with $d\geq 2$ fixed, we consider
the following $d\times d$ matrix-valued process
$X^{\beta}=\\{X_{i,j}^{\beta}(t);t\in\mathbb{R}_{+}^{N},1\leq i,j\leq d\\}$
whose entries are given by

$\displaystyle
X_{i,j}^{\beta}(t)=\begin{cases}\xi_{i,j}(t)+\iota\mathbf{1}_{[\beta=2]}\eta_{i,j}(t),&i<j;\\\
\sqrt{2}\xi_{i,i}(t),&i=j;\\\
\xi_{j,i}(t)-\iota\mathbf{1}_{[\beta=2]}\eta_{j,i}(t),&i>j.\end{cases}$ (1.11)

Clearly, for every $t\in\mathbb{R}_{+}^{N}$, $X^{\beta}(t)$ is a real
symmetric Gaussian matrix for $\beta=1$ and a complex Hermitian Gaussian
matrix for $\beta=2$. Moreover, the matrix Gaussian field $X^{\beta}$ extends
the matrix Brownian motion (1.1), matrix fractional Brownian motion (1.6) and
matrix Brownian sheet (1.9). In particular, $X^{1}(t)/\sqrt{C(t,t)}$ belongs
to the Gaussian orthogonal ensemble (GOE) and $X^{2}(t)/\sqrt{2C(t,t)}$
belongs to the Gaussian unitary ensemble (GUE), respectively. Let $A^{1}$ be a
real symmetric deterministic matrix and $A^{2}$ be a complex Hermitian
deterministic matrix. Let
$\\{\lambda_{1}^{\beta}(t),\cdots,\lambda_{d}^{\beta}(t)\\}$ is the set of
eigenvalue processes of

$\displaystyle Y^{\beta}(t)=A^{\beta}+T_{\beta}X^{\beta}(t)T_{\beta}^{*}$
(1.12)

for $\beta=1,2$, where $T_{\beta}$ is a $d\times d$ real ($\beta=1$) or
complex ($\beta=2$) matrix.

To our best of knowledge, the model (1.12) is studied in the literature only
when $T_{\beta}=I_{d}$ and $r=1$. Under certain conditions on the Gaussian
random field $\xi$, [Jaramillo2018] studied the probability of collision of at
least two eigenvalue processes of $Y^{\beta}$ using hitting probability of
Gaussian random field, and provided a sharp condition under which the
collision probability is zero. The results were extended to the collision of
at least $k$ eigenvalue processes in [Xiao2021]. Moreover, [Xiao2021]
characterized the size of the set

${\cal C}_{k}^{\beta}=\\{t\in
I:\,\lambda_{i_{1}}^{\beta}(t)=\cdots=\lambda_{i_{k}}^{\beta}(t)\mathrm{\ for\
some\ }\,1\leq i_{1}<\cdots<i_{k}\leq d\\}$

of $k$-collision times by computing its Hausdorff dimension. However, the two
papers does not provide any useful information on collision probability for
the critical dimension. The critical dimension case of collision problem was
solved in [Lee2023] for $T_{\beta}$ being identity matrix.

For $n\in\mathbb{N}$, we denote $[n]$ as the set of integers in $[1,n]$. Let
$\\{a_{j}\\}_{j\in J}$ be a collection of numbers with index set $J$, and
$J^{\prime}$ is a subset of $J$, then we denote $a_{J^{\prime}}=\\{a_{j}:j\in
J^{\prime}\\}$. For any set $J^{\prime}$, we write $|J^{\prime}|$ for the
number of distinct elements in $J^{\prime}$. Then $|a_{J^{\prime}}|=1$ means
that $a_{j}$ are the same for all $j\in J^{\prime}$. In other words, all
elements in $a_{J^{\prime}}$ collide. The first goal of the present paper is
to investigate the following probability of multiple collisions of eigenvalue
processes of

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\subset[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right),$

where $2\leq l_{1},\ldots,l_{r}\leq d$ satisfying $\sum_{j=1}^{r}l_{j}\leq d$.
For general matrix $T_{\beta}$, we provide the sufficient and necessary
conditions such that the probability is zero in Theorem 1.1 for the real case
($\beta=1$) and Theorem 1.2 for the complex case ($\beta=2$). When the
collision probability is positive, we determine the Hausdorff dimension of the
set of collision times

$\displaystyle{\cal C}_{l_{1},\ldots,l_{r}}^{\beta}=\left\\{t\in
I:\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\subset[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq
r\right\\}.$ (1.13)

The two theorems (Theorem 1.1 and Theorem 1.2) extend previous results by
considering a more general multiple collisions problem of eigenvalue processes
for the more comprehensive model (1.12). Our results are reduced to
[Jaramillo2018, Xiao2021, Lee2023] when $T_{\beta}$ is identity and $r=1$. The
major difficulty is to handle the eigenvectors associated with multiple
eigenvalues when computing the dimension of the eigenvector matrix in the
spectral decomposition. The argument of [Jaramillo2018, Xiao2021, Lee2023] is
based on the fact that, if there is only one multiple eigenvalue, then the
eigenspace associated with the multiple eigenvalue is the orthogonal
complement of the space spanned by all eigenvectors of single eigenvalues.
However, the argument no longer works if $r>1$, where there are different
multiple eigenvalues. See Remark 3.1 for more details. In this paper, we
design Lie groups such that the quotient manifold of the Lie groups we
designed is exactly the set of eigenvector matrices. This is the main novelty
in methodology of our paper.

For $d_{1}\leq d_{2}$, consider the following $d_{1}\times d_{2}$ matrix-
valued process $W^{\beta}=\\{W_{i,j}^{\beta}(t);t\in\mathbb{R}_{+}^{N},1\leq
i\leq d_{1},1\leq j\leq d_{2}\\}$ with entries
$W_{i,j}^{\beta}(t)=\xi_{i,j}(t)+\iota\mathbf{1}_{[\beta=2]}\eta_{i,j}(t)$.
Obviously, $W^{\beta}(t)W^{\beta}(t)^{*}/(\beta C(t,t))$, and
$W^{\beta}(t)^{*}W^{\beta}(t)/(\beta C(t,t))$ are Wishart matrices for
$\beta=1,2$. Let $B^{1}$ be a real deterministic $d_{1}\times d_{2}$ matrix
and $B^{2}$ be a complex deterministic $d_{1}\times d_{2}$ matrix. We consider
the following matrix

$\displaystyle Z^{\beta}(t)=B^{\beta}+T_{\beta}W^{\beta}(t)\tilde{T}_{\beta},$
(1.14)

where $T_{\beta}$ and $\tilde{T}_{\beta}$ are $d_{1}\times d_{1}$ and
$d_{2}\times d_{2}$ real ($\beta=1$) or complex ($\beta=2$) matrices
respectively. Let $\\{s_{1}^{\beta}(t),\ldots,s_{d_{1}}^{\beta}(t)\\}$ be the
set of non-trivial singular value processes of $Z^{\beta}(t)$. The second goal
of the present paper is to examine when the probability

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|s_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d_{1}]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)$

is zero. We provide the sufficient and necessary conditions for the
probability being zero in Theorem 1.3 for the real case and in Theorem 1.4 for
the complex case. We also compute the Hausdorff dimension of the set of
collision times

$\displaystyle\widetilde{\cal C}_{l_{1},\ldots,l_{r}}^{\beta}=\left\\{t\in
I:\left|s_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\subset[d_{1}]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq
r\right\\}.$ (1.15)

We remark that the set of eigenvalue processes of
$Z^{\beta}(t)Z^{\beta}(t)^{*}$ is
$\\{s_{1}^{\beta}(t)^{2},\ldots,s_{d_{1}}^{\beta}(t)^{2}\\}$, while the set of
eigenvalue processes of $Z^{\beta}(t)^{*}Z^{\beta}(t)$ is
$\\{s_{1}^{\beta}(t)^{2},\ldots,s_{d_{1}}^{\beta}(t)^{2},0,\ldots,0\\}$. Thus,
the knowledge of collision of singular value processes of $Z^{\beta}(t)$ is
equivalent to that of eigenvalue processes of $Z^{\beta}(t)Z^{\beta}(t)^{*}$
or $Z^{\beta}(t)^{*}Z^{\beta}(t)$. Thus, our results extend the non-collision
property of eigenvalue processes of Wishart process and fractional Wishart
process.

To our best knowledge, the two theorems (Theorem 1.3 and Theorem 1.4) are the
first results on the collision probability of singular value processes of
matrix Gaussian random field with high dimensional time parameter. To
establish the two theorems, we shall consider the singular value
decomposition. The main difficulty arises from handling the singular vectors
associated with multiple singular values, especially for the trivial singular
value 0, noting that there are $d_{2}-d_{1}$ right singular vectors that
associated with the trivial singular value 0. To address the difficulty of
multiple singular values, we identify the matrices of left singular vectors
and right singular vectors as the quotient manifold of well-designed Lie
groups.

The rest of the paper is organized as follows. We present the assumptions on
the Gaussian random field $\xi$ in Subsection 1.2, introduce our main theorems
in Subsection 1.3 and fix some notations in Subsection 1.4. We present some
preliminaries that will be used in the proofs of main theorems in Section 2.
Then we develop the proofs of Theorem 1.1, Theorem 1.2, Theorem 1.3 and
Theorem 1.4 in Subsection 3.1, Subsection 3.2, Subsection 4.1 and Subsection
4.2 respectively.

###  1.2 Assumptions

We first recall the following definition of an $\mathbb{R}^{d}$-valued
Gaussian noise on $\mathbb{R}_{+}$:

######  Definition 1.

Let $\nu$ be a Borel measure on $\mathbb{R}_{+}$, and let $A\mapsto W(A)$ be a
set function defined on $\mathcal{B}(\mathbb{R}_{+})$ with values in
$L^{2}(\Omega,\mathcal{F},\mathbb{P};\mathbb{R}^{d})$ such that for each $A$,
$W(A)$ is a centered normal random vector with values in $\mathbb{R}^{d}$ and
covariance matrix $\nu(A)I_{d}$. Assume that $W(A\cup B)=W(A)+W(B)$ a.s., and
$W(A)$ and $W(B)$ are independent whenever $A\cap B=\emptyset$. Then the set
function $A\mapsto W(A)$ is called an $\mathbb{R}^{d}$-valued Gaussian noise
with control measure $\nu$.

Recall the compact interval $I$ given by (1.10), we write
$I^{(\epsilon)}=\prod_{j=1}^{N}(a_{j}-\epsilon,b_{j}+\epsilon)$ for any
$\epsilon>0$. Then we can impose the assumptions on the Gaussian random field
$\xi=\\{\xi(t):t\in\mathbb{R}_{+}^{N}\\}$. Let $0<H_{1}\leq\ldots\leq H_{N}<1$
be a non-decreasing sequence in $(0,1)$. We assume that the following
conditions hold:

  1. (A1)

There is a Gaussian random field
$\\{W(A,t):A\in\mathcal{B}(\mathbb{R}_{+}),t\in\mathbb{R}^{N}\\}$ and
$\epsilon_{0}>0$ satisfying the following two conditions:

    1. (c1)

For all $t\in I^{(\epsilon_{0})}$, $A\mapsto W(A,t)$ is an $\mathbb{R}$-valued
Gaussian noise with a control measure $\nu_{t}$ such that
$W(\mathbb{R}_{+},t)=\xi(t)$ and when $A\cap B=\emptyset$, $W(A,\cdot)$ and
$W(B,\cdot)$ are independent.

    2. (c2)

There exist constants $a_{0}\geq 0$, $c_{0}>0$, such that for all $a_{0}\leq
c<d\leq+\infty$ and all $s:=(s_{1},\dots,s_{N}),t:=(t_{1},\dots,t_{N})\in
I^{(\epsilon_{0})}$,

$\displaystyle\big{\|}W([c,d),s)-X(s)-W([c,d),t)+X(t)\big{\|}_{L^{2}}\leq
c_{0}\left[\sum_{j=1}^{N}c^{H_{j}^{-1}-1}|s_{j}-t_{j}|+d^{-1}\right],$

and

$\displaystyle\big{\|}W([0,a_{0}),s)-W([0,a_{0}),t)\big{\|}_{L^{2}}\leq
c_{0}\sum_{j=1}^{N}|s_{j}-t_{j}|.$

  2. (A2)

There exists a constant $c_{1}>0$, such that $\|\xi(t)\|_{L^{2}}\geq c_{1}$
for all $t\in I^{(\epsilon_{0})}$.

  3. (A3)

There exists a constant ${\rho_{0}}>0$ with the following property. For $t\in
I$, there exist $t^{\prime}=t^{\prime}(t)\in I^{(\epsilon_{0})}$,
$\delta_{j}=\delta_{j}(t)\in(H_{j},1]$ for $1\leq i\leq N$, and
$c_{2}=c_{2}(t)>0$, such that

$\displaystyle\Big{|}\mathbb{E}\left[\xi(t^{\prime})\big{(}\xi(s)-\xi(\bar{s})\big{)}\right]\Big{|}\leq
c_{2}\sum_{j=1}^{N}|s_{j}-\bar{s}_{j}|^{\delta_{j}},$

for all $1\leq i\leq N$ and all $s,\bar{s}\in I^{(\epsilon_{0})}$ with
$\max\\{\sum_{j=1}^{N}|s_{j}-t_{j}|^{H_{j}},\sum_{j=1}^{N}|\bar{s}_{j}-t_{j}|^{H_{j}}\\}\leq
2{\rho_{0}}$.

  4. (A4)

There exists positive and finite constant $c_{3}$, such that

$\displaystyle\left\|\xi(s)-\xi(t)\right\|_{L^{2}}\geq
c_{3}\sum_{j=1}^{N}|s_{j}-t_{j}|^{H_{j}}$

for all $s=(s_{1},\ldots,s_{N}),t=(t_{1},\ldots,t_{N})\in I$.

  5. (A5)

There exists a positive constant $c_{4}$ such that for all $s,t\in I$,

$\displaystyle\mathrm{Var}\left[\xi(t)|\xi(s)\right]\geq
c_{4}\sum_{j=1}^{N}|s_{j}-t_{j}|^{2H_{j}},$

where $\mathrm{Var}\left[\xi(t)|\xi(s)\right]$ denotes the conditional
variance of $\xi(t)$ given $\xi(s)$.

######  Remark 1.1.

By [Xiao2009]*page 256, the conditional variance of centered Gaussian vector
has the following formula:

$\displaystyle\mathrm{Var}\left(U\big{|}V\right)=\dfrac{\left(\rho^{2}_{U,V}-\left(\sigma_{U}-\sigma_{V}\right)^{2}\right)\left(\left(\sigma_{U}+\sigma_{V}\right)^{2}-\rho^{2}_{U,V}\right)}{4\sigma_{V}^{2}},$

where $\rho^{2}_{U,V}=\mathbb{E}[(U-V)^{2}]$,
$\sigma_{U}^{2}=\mathbb{E}[U^{2}]$ and $\sigma_{V}^{2}=\mathbb{E}[V^{2}]$.

######  Remark 1.2.

By [dalang2017polarity]*Proposition 2.2, Assumption (A1) implies that there
exists a constant $C>0$, such that

$\displaystyle\left\|\xi(s)-\xi(t)\right\|_{L^{2}}\leq
C\sum_{j=1}^{N}|s_{j}-t_{j}|^{H_{j}}.$

for all $s,t\in I^{(\epsilon_{0})}$. See also [Lee2023]*Lemma 2.2.

######  Remark 1.3.

Assumptions (A1) \- (A3) are slightly stronger than the assumptions in
[Lee2023], and will be used for the critical dimension of hitting probability.
By Remark 1.2, Assumption (A1), (A2), (A4) and (A5) are slightly stronger than
the assumptions in [Jaramillo2018] and [Xiao2021], and are required for the
non-critical dimension of hitting probability.

######  Remark 1.4.

The Assumptions (A1) \- (A5) include a large class of Gaussian random fields
such as multi-parameter fractional Brownian motions, fractional Brownian
sheets, solutions to stochastic heat equations. We refer the interested
readers to [xiao2009sample, Xiao2021, Lee2023] for more details.

###  1.3 Main results

Let $\xi=\\{\xi(t):t\in\mathbb{R}_{+}^{N}\\}$ be the Gaussian random field
satisfying Assumptions (A1) \- (A5), and
$\\{\xi_{i,j},\eta_{i,j}:i,j\in\mathbb{N}\\}$ be a family of independent
copies of $\xi$. We assume that the matrices $T_{\beta},\tilde{T}_{\beta}$ are
invertible.

The following is the first results of this paper, which extends the previous
results ([Jaramillo2018, Xiao2021, Lee2023]) to existence of different
multiple eigenvalue processes. For the real-valued case, we have the following
theorem.

######  Theorem 1.1.

Let $Y^{\beta}$ ($\beta=1$) be the matrix-valued process defined by (1.12)
with eigenvalues $\\{\lambda_{1}^{\beta}(t),\dots,\lambda_{d}^{\beta}(t)\\}$.
Then for any $l_{1},\ldots,l_{r}\in\\{2,\ldots,d\\}$ satisfying
$\sum_{j=1}^{r}l_{j}\leq d$, the following statements hold:

  1. (i)

If $\sum_{j=1}^{N}\frac{1}{H_{j}}\leq\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)/2$, then

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)=0.$

  2. (ii)

If $\sum_{j=1}^{N}\frac{1}{H_{j}}>\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)/2$, then

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)>0.$

  3. (iii)

If $\sum_{j=1}^{N}\frac{1}{H_{j}}>\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)/2$, then
with positive probability, the Hausdorff dimension of the set ${\cal
C}_{l_{1},\ldots,l_{r}}^{\beta}$ of collision times given in (1.13) is

$\begin{split}\dim_{{}_{\rm H}}{\cal
C}_{l_{1},\ldots,l_{r}}^{\beta}&=\min_{1\leq\ell\leq
N}\bigg{\\{}\sum_{j=1}^{\ell}\frac{H_{\ell}}{H_{j}}+N-\ell-\dfrac{H_{\ell}}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)\bigg{\\}}\\\
&=\sum_{j=1}^{\ell_{0}}\frac{H_{\ell_{0}}}{H_{j}}+N-\ell_{0}-\dfrac{H_{\ell_{0}}}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2),\end{split}$
(1.16)

where $\ell_{0}$ is the smallest $\ell$ such that
$\sum_{j=1}^{\ell}\frac{1}{H_{j}}>\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)/2$, i.e.,
$\sum_{j=1}^{\ell_{0}-1}\frac{1}{H_{j}}\leq\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)/2<\sum_{j=1}^{\ell_{0}}\frac{1}{H_{j}}$
with the convention $\sum_{j=1}^{0}\frac{1}{H_{j}}:=0$.

To illustrate this theorem, we consider the special case of a symmetric
matrix-valued process $X^{\beta}$ associated with fractional Brownian motion
$B^{H}$, noting that $B^{H}$ satisfies Assumptions (A1) - (A5).

######  Corollary 1.1.

Let $\beta=1$ and $Y^{\beta}$ be a matrix-valued process given in (1.12)
associated with fractional Brownian motion $B^{H}$. Then for any
$l_{1},\ldots,l_{r}\in\\{2,\ldots,d\\}$ satisfying $\sum_{j=1}^{r}l_{j}\leq
d$, the following hold:

  1. (i)

If $N\leq H\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)/2$, then

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)=0.$

  2. (ii)

If $N>H\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)/2$, then

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)>0.$

  3. (iii)

If $N>H\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)/2$, then with positive probability,

$\dim_{{}_{\rm H}}{\cal
C}_{l_{1},\ldots,l_{r}}^{\beta}=N-\dfrac{H}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2).$

The following theorem is the complex analog of Theorem 1.1.

######  Theorem 1.2.

Let $Y^{\beta}$ ($\beta=2$) be the matrix-valued process defined by (1.12)
with eigenvalues $\\{\lambda_{1}^{\beta}(t),\dots,\lambda_{d}^{\beta}(t)\\}$.
Then for any $l_{1},\ldots,l_{r}\in\\{2,\ldots,d\\}$ satisfying
$\sum_{j=1}^{r}l_{j}\leq d$, the following statements hold:

  1. (i)

If $\sum_{j=1}^{N}\frac{1}{H_{j}}\leq\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1)$, then

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)=0.$

  2. (ii)

If $\sum_{j=1}^{N}\frac{1}{H_{j}}>\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1)$, then

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)>0.$

  3. (iii)

If $\sum_{j=1}^{N}\frac{1}{H_{j}}>\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1)$, then with
positive probability, the Hausdorff dimension of the set ${\cal
C}_{k}^{\beta}$ of collision times given in (1.13) is

$\begin{split}\dim_{{}_{\rm H}}{\cal
C}_{l_{1},\ldots,l_{r}}^{\beta}&=\min_{1\leq\ell\leq
N}\bigg{\\{}\sum_{j=1}^{\ell}\frac{H_{\ell}}{H_{j}}+N-\ell-
H_{\ell}\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1)\bigg{\\}}\\\
&=\sum_{j=1}^{\ell_{0}}\frac{H_{\ell_{0}}}{H_{j}}+N-\ell_{0}-H_{\ell_{0}}\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1),\end{split}$

where $\ell_{0}$ is the smallest $\ell$ such that
$\sum_{j=1}^{\ell}\frac{1}{H_{j}}>\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1)$.

Similarly, we have the following corollary where the Gaussian random field
$\xi$ is reduced to the fractional Brownian motion $B^{H}$.

######  Corollary 1.2.

Let $\beta=2$ and $Y^{\beta}$ be a matrix-valued process given in (1.12)
associated with fractional Brownian motion $B^{H}$. Then for any
$l_{1},\ldots,l_{r}\in\\{2,\ldots,d\\}$ satisfying $\sum_{j=1}^{r}l_{j}\leq
d$, the following hold:

  1. (i)

If $N\leq H\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1)$, then

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)=0.$

  2. (ii)

If $N>H\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1)$, then

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)>0.$

  3. (iii)

If $N>H\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1)$, then with positive probability,

$\dim_{{}_{\rm H}}{\cal
C}_{l_{1},\ldots,l_{r}}^{\beta}=N-H\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1).$

Next, we will turn to the collision of singular value processes of
$Z^{\beta}(t)$. Recall that $Z^{\beta}(t)$ is a $d_{1}\times d_{2}$ matrix
given by (1.14) with $d_{1}\leq d_{2}$. Let’s start with the real case.

######  Theorem 1.3.

Let $Z^{\beta}$ ($\beta=1$) be the matrix-valued process defined by (1.14)
with non-trivial singular values
$\left\\{s_{1}^{\beta}(t),\dots,s_{d_{1}}^{\beta}(t)\right\\}$. Then for any
$l_{1},\ldots,l_{r}\in\\{2,\ldots,d_{1}\\}$ satisfying
$\sum_{j=1}^{r}l_{j}\leq d_{1}$, the following statements hold:

  1. (i)

If
$\sum_{j=1}^{N}\frac{1}{H_{j}}\leq\frac{1}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)$,
then

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|s_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d_{1}]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq
r\right)=0.$

  2. (ii)

If
$\sum_{j=1}^{N}\frac{1}{H_{j}}>\frac{1}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)$,
then

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|s_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d_{1}]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq
r\right)>0.$

  3. (iii)

If
$\sum_{j=1}^{N}\frac{1}{H_{j}}>\frac{1}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)$,
then with positive probability, the Hausdorff dimension of the set
$\widetilde{\cal C}_{l_{1},\ldots,l_{r}}^{\beta}$ of collision times given in
(1.13) is

$\begin{split}\dim_{{}_{\rm H}}\widetilde{\cal
C}_{l_{1},\ldots,l_{r}}^{\beta}&=\min_{1\leq\ell\leq
N}\bigg{\\{}\sum_{j=1}^{\ell}\frac{H_{\ell}}{H_{j}}+N-\ell-\frac{H_{\ell}}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)\bigg{\\}}\\\
&=\sum_{j=1}^{\ell_{0}}\dfrac{H_{\ell_{0}}}{H_{j}}+N-\ell_{0}-\dfrac{H_{\ell_{0}}}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2),\end{split}$
(1.17)

where $\ell_{0}$ is the smallest $\ell$ such that
$\sum_{j=1}^{\ell}\frac{1}{H_{j}}>\frac{1}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)$.

If the Gaussian random field $\xi$ is reduced to the fractional Brownian
motion, we have the following corollary.

######  Corollary 1.3.

Let $\beta=1$ and $Z^{\beta}$ be a matrix-valued process given in (1.14)
associated with fractional Brownian motion $B^{H}$. Then for any
$l_{1},\ldots,l_{r}\in\\{2,\ldots,d_{1}\\}$ satisfying
$\sum_{j=1}^{r}l_{j}\leq d_{1}$, the following hold:

  1. (i)

If $N<\frac{H}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)$, then

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)=0.$

  2. (ii)

If $N>\frac{H}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)$, then

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)>0.$

  3. (iii)

If $N>\frac{H}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)$, then with positive
probability,

$\dim_{{}_{\rm H}}\widetilde{\cal
C}_{l_{1},\ldots,l_{r}}^{\beta}=N-\dfrac{H}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2).$

The next theorem is the complex version of Theorem 1.3.

######  Theorem 1.4.

Let $Z^{\beta}$ ($\beta=2$) be the matrix-valued process defined by (1.14)
with non-trivial singular values
$\left\\{s_{1}^{\beta}(t),\dots,s_{d_{1}}^{\beta}(t)\right\\}$. Then for any
$l_{1},\ldots,l_{r}\in\\{2,\ldots,d_{1}\\}$ satisfying
$\sum_{j=1}^{r}l_{j}\leq d_{1}$, the following statements hold:

  1. (i)

If $\sum_{j=1}^{N}\frac{1}{H_{j}}\leq\sum_{j=1}^{r}(l_{j}-1)(l_{j}+1)$, then

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|s_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d_{1}]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq
r\right)=0.$

  2. (ii)

If $\sum_{j=1}^{N}\frac{1}{H_{j}}>\sum_{j=1}^{r}(l_{j}-1)(l_{j}+1)$, then

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|s_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d_{1}]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq
r\right)>0.$

  3. (iii)

If $\sum_{j=1}^{N}\frac{1}{H_{j}}>\sum_{j=1}^{r}(l_{j}-1)(l_{j}+1)$, then with
positive probability, the Hausdorff dimension of the set $\widetilde{\cal
C}_{l_{1},\ldots,l_{r}}^{\beta}$ of collision times given in (1.13) is

$\begin{split}\dim_{{}_{\rm H}}\widetilde{\cal
C}_{l_{1},\ldots,l_{r}}^{\beta}&=\min_{1\leq\ell\leq
N}\left\\{\sum_{j=1}^{\ell}\frac{H_{\ell}}{H_{j}}+N-\ell-
H_{\ell}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+1)\right\\}\\\
&=\sum_{j=1}^{\ell_{0}}\frac{H_{\ell_{0}}}{H_{j}}+N-\ell_{0}-H_{\ell_{0}}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+1),\end{split}$

where $\ell_{0}$ is the smallest $\ell$ such that
$\sum_{j=1}^{\ell}\frac{1}{H_{j}}>\sum_{j=1}^{r}(l_{j}-1)(l_{j}+1)$.

If the associated Gaussian random field $\xi$ is a fractional Brownian motion
$B^{H}$, then Theorem 1.4 reduced to the following corollary.

######  Corollary 1.4.

Let $\beta=2$ and $Z^{\beta}$ be a matrix-valued process given in (1.14)
associated with fractional Brownian motion $B^{H}$. Then for any
$l_{1},\ldots,l_{r}\in\\{2,\ldots,d_{1}\\}$ satisfying
$\sum_{j=1}^{r}l_{j}\leq d_{1}$, the following hold:

  1. (i)

If $N\leq H\sum_{j=1}^{r}(l_{j}-1)(l_{j}+1)$, then

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)=0.$

  2. (ii)

If $N>H\sum_{j=1}^{r}(l_{j}-1)(l_{j}+1)$, then

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)>0.$

  3. (iii)

If $N>H\sum_{j=1}^{r}(l_{j}-1)(l_{j}+1)$, then with positive probability,

$\dim_{{}_{\rm H}}\widetilde{\cal
C}_{l_{1},\ldots,l_{r}}^{\beta}=N-H\sum_{j=1}^{r}(l_{j}-1)(l_{j}+1).$

###  1.4 Notations

In this subsection, we introduce some notations on matrices that will be used
in the proofs.

For $d,d_{1},d_{2}\in\mathbb{N}$, we denote by ${\mathbf{D}}(d_{1}\times
d_{2})$ the set of diagonal real $d_{1}\times d_{2}$ matrices (the set of
matrices whose $(i,j)$ entry vanishes for $i\neq j$) and
${\mathbf{D}}(d)={\mathbf{D}}(d\times d)$. For a matrix $A$, denote by $A^{*}$
the conjugate transpose of $A$. We also use the notation $A_{*,j}$ for the
$j$th column of $A$. If $A$ is a square matrix, then we denote by
$\mathbf{Spec}(A)$ the spectrum of $A$, i.e. the set of eigenvalues of $A$. We
write $\mathcal{E}^{A}_{\lambda}$ for the eigenspace associated with
$\lambda\in\mathbf{Spec}(A)$.

We denote by $\mathbb{R}^{d_{1}\times d_{2}}$ the set of real $d_{1}\times
d_{2}$ matrix, and identify its element

$\displaystyle\tilde{x}=\left(\begin{matrix}\tilde{x}_{11}&\tilde{x}_{12}&\cdots&\tilde{x}_{1d_{2}}\\\
\tilde{x}_{21}&\tilde{x}_{22}&\cdots&\tilde{x}_{2d_{2}}\\\
\ldots&\ldots&\ldots&\ldots\\\
\tilde{x}_{d_{1}1}&\tilde{x}_{d_{1}2}&\cdots&\tilde{x}_{d_{1}d_{2}}\end{matrix}\right)$

as the row vector

$\displaystyle\left(\tilde{x}_{11},\ldots,\tilde{x}_{1d_{2}},\tilde{x}_{21},\ldots,\tilde{x}_{2d_{2}},\ldots,\tilde{x}_{d_{1}1},\ldots,\tilde{x}_{d_{1}d_{2}}\right)$

in $\mathbb{R}^{d_{1}d_{2}}$. Similarly, we use $\mathbb{C}^{d_{1}\times
d_{2}}$ for the set of complex $d_{1}\times d_{2}$ matrix, and its elements
can be identify as row vectors in $\mathbb{R}^{2d_{1}d_{2}}$ by considering
the real part and imaginary part.

We denote by $\mathbf{S}(d)$ and $\mathbf{H}(d)$ the set of real symmetric
$d\times d$ matrices and the set of complex Hermitian $d\times d$ matrices,
respectively. A symmetric matrix

$\displaystyle\tilde{x}=\left(\begin{matrix}\tilde{x}_{11}&\tilde{x}_{12}&\cdots&\tilde{x}_{1d}\\\
\tilde{x}_{12}&\tilde{x}_{22}&\cdots&\tilde{x}_{2d}\\\
\ldots&\ldots&\ldots&\ldots\\\
\tilde{x}_{1d}&\tilde{x}_{2d}&\cdots&\tilde{x}_{dd}\end{matrix}\right)$

can be viewed as a row vector

$\displaystyle\left(\tilde{x}_{11},\ldots,\tilde{x}_{1d},\tilde{x}_{22},\ldots,\tilde{x}_{2d},\ldots,\tilde{x}_{dd}\right)$

in $\mathbb{R}^{d(d+1)/2}$. In a similar way, a Hermitian matrix

$\displaystyle\tilde{x}=\left(\begin{matrix}\tilde{x}_{11}&\tilde{x}_{12}&\cdots&\tilde{x}_{1d}\\\
\overline{\tilde{x}_{12}}&\tilde{x}_{22}&\cdots&\tilde{x}_{2d}\\\
\ldots&\ldots&\ldots&\ldots\\\
\overline{\tilde{x}_{1d}}&\overline{\tilde{x}_{2d}}&\cdots&\tilde{x}_{dd}\end{matrix}\right)$

can be understood as a row vector

$\displaystyle\Big{(}\tilde{x}_{11},\mathrm{Re}(\tilde{x}_{12}),\ldots,\mathrm{Re}(\tilde{x}_{1d}),\tilde{x}_{22},\mathrm{Re}(\tilde{x}_{23}),\ldots,\mathrm{Re}(\tilde{x}_{2d}),\ldots,\tilde{x}_{dd},$
$\displaystyle~{}~{}~{}~{}~{}\qquad\qquad\qquad\quad\mathrm{Im}(\tilde{x}_{12}),\ldots,\mathrm{Im}(\tilde{x}_{1d}),\mathrm{Im}(\tilde{x}_{23}),\ldots,\mathrm{Im}(\tilde{x}_{(d-1)d})\Big{)}.$

in $\mathbb{R}^{d^{2}}$.

By the identification above, we abuse the notation, namely, for a vector $x$
in $\mathbb{R}^{d(d+1)/2}$, $\mathbb{R}^{d^{2}}$, $\mathbb{R}^{d_{1}d_{2}}$ or
$\mathbb{R}^{2d_{1}d_{2}}$, it also means the corresponding matrix in
$\mathbf{S}(d)$, $\mathbf{H}(d)$, $\mathbb{R}^{d_{1}\times d_{2}}$ or
$\mathbb{C}^{d_{1}\times d_{2}}$ respectively.

For a vector $x$ in $\mathbb{R}^{d(d+1)/2}$ or $\mathbb{R}^{d^{2}}$, let
$E_{i}(x)$ be the $i$th smallest eigenvalue of $x$. Then $E_{i}(x)$ is a
continuous function of $x$ for each $i\in\\{1,\dots,d\\}$, noting that
$(E_{1}(x),\dots,E_{d}(x))$ are ordered roots of the characteristic polynomial
of $x$. For a vector $x$ in $\mathbb{R}^{d_{1}d_{2}}$ or
$\mathbb{R}^{2d_{1}d_{2}}$, let $S_{i}(x)$ be the $i$th smallest singular
value of $x$. Then $S_{i}(x)$ is a continuous function of $x$ for each
$i\in\\{1,\dots,d_{1}\\}$, noting that $(S_{1}(x)^{2},\dots,S_{d_{1}}(x)^{2})$
are ordered eigenvalues of $xx^{*}$.

For $r\in\mathbb{N}$, $l_{1},\ldots,l_{r}\in\\{2,\ldots,d\\}$ satisfying
$\sum_{j=1}^{r}l_{j}\leq d$, we define

$\displaystyle{\mathbf{S}}(d;l_{1},\ldots,l_{r})=\\{x\in{\mathbf{S}}(d):\
|E_{J_{j}}|=1,\mathrm{\ for\ some\ disjoint\ }J_{j}\subset[d]\mathrm{\ with\
}|J_{j}|=l_{j},1\leq j\leq r\\}$ (1.18)

and

$\displaystyle{\mathbf{H}}(d;l_{1},\ldots,l_{r})=\\{x\in{\mathbf{H}}(d):\
|E_{J_{j}}|=1,\mathrm{\ for\ some\ disjoint\ }J_{j}\subset[d]\mathrm{\ with\
}|J_{j}|=l_{j},1\leq j\leq r\\}.$ (1.19)

For $d_{1}\leq d_{2}$, if $l_{1},\ldots,l_{r}\in\\{2,\ldots,d_{1}\\}$
satisfying $\sum_{j=1}^{r}l_{j}\leq d_{1}$, we define

$\displaystyle{\mathbf{M}}_{\mathbb{R}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})=$ $\displaystyle\\{x\in\mathbb{R}^{d_{1}\times
d_{2}}:\ |S_{J_{j}}|=1,$ $\displaystyle\qquad\qquad\mathrm{for\ some\
disjoint\ }J_{j}\subset[d_{1}]\mathrm{\ with\ }|J_{j}|=l_{j},1\leq j\leq r\\}$
(1.20)

and

$\displaystyle{\mathbf{M}}_{\mathbb{C}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})=$ $\displaystyle\\{x\in\mathbb{C}^{d_{1}\times
d_{2}}:\ |S_{J_{j}}|=1,$ $\displaystyle\qquad\qquad\mathrm{for\ some\
disjoint\ }J_{j}\subset[d_{1}]\mathrm{\ with\ }|J_{j}|=l_{j},1\leq j\leq
r\\}.$ (1.21)

For a vector space $\mathbb{R}^{d}$ or $\mathbb{C}^{d}$, let $\|\cdot\|$ be
the Euclidean norm and $\langle\cdot,\cdot\rangle$ be the corresponding inner
product. For a metric space $X$, we denote by $\mathfrak{B}_{r}(x)$ the open
ball centered at $x\in X$ with radius $r$. For $A\in\mathbb{C}^{m\times n}$,
we define the Frobenius norm,

$\|A\|=\bigg{(}\sum_{i=1}^{m}\sum_{j=1}^{n}|A_{ij}|^{2}\bigg{)}^{1/2}.$ (1.22)

Thus $\|A\|$ is just the Euclidean norm of $A$, if we consider $A$ as a vector
of size $mn$.

##  2 Preliminaries

###  2.1 Hausdorff dimension and capacity

Firstly, we briefly recall the definitions of Hausdorff measure, Hausdorff
dimension, and Bessel-Riesz capacity as well as their relationship that are
used in the present paper. We refer to [Falconer2014, mattila1999] for more
details.

Let $n\in\mathbb{N}$ be fixed and $q>0$ be a constant. For any subset
$A\subseteq\mathbb{R}^{n}$, the $q$-dimensional Hausdorff measure of $A$ is
defined by

$\mathcal{H}_{q}(A)=\lim_{\varepsilon\to
0}\inf\bigg{\\{}\sum_{i}(2r_{i})^{q}:\,A\subseteq\bigcup_{i=1}^{\infty}\mathfrak{B}_{r_{i}}(x_{i}),\
r_{i}<\varepsilon\bigg{\\}}.$

We use the convention that $\mathcal{H}_{q}(A)=1$ when $q\leq 0$. It is known
that $\mathcal{H}_{q}(\cdot)$ is a metric outer measure and every Borel set in
$\mathbb{R}^{n}$ is $\mathcal{H}_{q}$-measurable. The Hausdorff dimension
$\dim_{{}_{\rm H}}(A)$ defined by

$\displaystyle\dim_{{}_{\rm H}}(A)=\inf\\{q>0\ |\
\mathcal{H}_{q}(A)=0\\}=\sup\\{q>0\ |\ \mathcal{H}_{q}(A)>0\\}.$

The Bessel-Riesz capacity of order $q$ of $A$ is defined by

$\displaystyle\mathcal{C}_{q}(A)=\left(\inf_{\mu\in\mathcal{P}(A)}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}f_{q}(\|x-y\|)\mu(dx)\mu(dy)\right)^{-1},$

where $\mathcal{P}(A)$ is the family of probability measures supported in $A$,
and the function $f_{q}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is given by

$\displaystyle f_{q}(r)=\begin{cases}r^{-q},&q>0;\\\
\ln\left(\dfrac{e}{r\wedge 1}\right),&q=0;\\\ 1,&q<0.\end{cases}$

Hausdorff dimension and Bessel-Riesz capacity are related by the following
Frostman theorem:

$\displaystyle\dim_{{}_{\rm H}}(A)=\inf\\{q>0\ |\
\mathcal{C}_{q}(A)=0\\}=\sup\\{q>0\ |\ \mathcal{C}_{q}(A)>0\\}.$

The following lemma gives the estimate of Hausdorff measure of the image of a
mapping.

######  Lemma 2.1.

Let $\mathcal{L}$ be a bijective mapping from $\mathbb{R}^{n}$ to
$\mathbb{R}^{n}$. Assume that there exists a positive constant
$c_{\mathcal{L}}$, such that

$\displaystyle\mathcal{L}^{-1}\left(\mathcal{B}_{r}(x)\right)\subseteq\mathcal{B}_{c_{\mathcal{L}}r}(\mathcal{L}^{-1}(x)),\quad\forall
x\in\mathbb{R}^{n},r\in\mathbb{R}_{+}.$ (2.1)

Then for any $A\subseteq\mathbb{R}^{n}$, we have

$\displaystyle\mathcal{H}_{q}\left(\mathcal{L}(A)\right)\leq
c_{\mathcal{L}}^{-q}\mathcal{H}_{q}\left(A\right),\quad\forall q>0.$

###### Proof.

By definition, for $q>0$,

$\displaystyle\mathcal{H}_{q}\left(\mathcal{L}(A)\right)$
$\displaystyle=\lim_{\varepsilon\to
0}\inf\bigg{\\{}\sum_{i}(2r_{i})^{q}:\,\mathcal{L}(A)\subseteq\bigcup_{i=1}^{\infty}\mathfrak{B}_{r_{i}}(x_{i}),\
r_{i}<\varepsilon\bigg{\\}}$ $\displaystyle\leq\lim_{\varepsilon\to
0}\inf\bigg{\\{}\sum_{i}(2r_{i})^{q}:\,A\subseteq\bigcup_{i=1}^{\infty}\mathfrak{B}_{c_{\mathcal{L}}r_{i}}\left(\mathcal{L}^{-1}(x_{i})\right),\
r_{i}<\varepsilon\bigg{\\}}$
$\displaystyle=c_{\mathcal{L}}^{-q}\lim_{\epsilon\to
0}\inf\bigg{\\{}\sum_{i}(2s_{i})^{q}:\,A\subseteq\bigcup_{i=1}^{\infty}\mathfrak{B}_{s_{i}}\left(y_{i}\right),\
s_{i}<\epsilon\bigg{\\}}$
$\displaystyle=c_{\mathcal{L}}^{-q}\mathcal{H}_{q}\left(A\right),$

where we change of variables $y_{i}=\mathcal{L}^{-1}(x_{i})$,
$s_{i}=c_{\mathcal{L}}r_{i}$ and $\epsilon=c_{\mathcal{L}}\varepsilon$. ∎

As a consequence, we have the following relationship of Hausdorff dimension of
any set $A$ and its image under $\mathcal{L}$.

######  Corollary 2.1.

Let $\mathcal{L}$ be a bijective mapping from $\mathbb{R}^{n}$ to
$\mathbb{R}^{n}$ satisfying (2.1), then for any $A\subseteq\mathbb{R}^{n}$,

$\displaystyle\dim_{{}_{\rm H}}(A)\geq\dim_{{}_{\rm
H}}\left(\mathcal{L}(A)\right).$ (2.2)

Moreover, if both $\mathcal{L}$ and $\mathcal{L}^{-1}$ satisfy (2.1), then

$\displaystyle\dim_{{}_{\rm H}}(A)=\dim_{{}_{\rm
H}}\left(\mathcal{L}(A)\right).$ (2.3)

###### Proof.

For any $q>\dim_{{}_{\rm H}}(A)$, we have $\mathcal{H}_{q}(A)=0$. Thus, by
Lemma 2.1, $\mathcal{H}_{q}(\mathcal{L}(A))=0$. Thus, by definition, we get
$\dim_{{}_{\rm H}}(\mathcal{L}(A))\leq q$. Then the proof of (2.2) is
concluded by setting $q\to\dim_{{}_{\rm H}}(A)$.

Applying (2.2) for the mapping $\mathcal{L}^{-1}$ and the set
$\mathcal{L}(A)$, we obtain

$\displaystyle\dim_{{}_{\rm H}}(\mathcal{L}(A))\geq\dim_{{}_{\rm
H}}\left(\mathcal{L}^{-1}(\mathcal{L}(A))\right)=\dim_{{}_{\rm H}}(A).$

∎

The following lemma helps to verify the assumption in Corollary 2.1 for a
class of matrix linear transformation $\mathcal{L}$.

######  Lemma 2.2.

Let $T,\tilde{T}$ be real or complex invertible matrices of dimension $d_{1}$
and $d_{2}$ respectively. Then the mapping
$\mathcal{L}:\mathbb{R}^{d_{1}\times d_{2}}\to\mathbb{R}^{d_{1}\times d_{2}}$
or $\mathcal{L}:\mathbb{C}^{d_{1}\times d_{2}}\to\mathbb{C}^{d_{1}\times
d_{2}}$ given by $\mathcal{L}(X)=TX\tilde{T}$ is well-defined, bijective with
inverse $\mathcal{L}^{-1}=T^{-1}X\tilde{T}^{-1}$. Moreover, both $\mathcal{L}$
and $\mathcal{L}^{-1}$ satisfy (2.1).

###### Proof.

By direct computation, $\mathcal{L}$ is well-defined and invertible. In the
following, we will prove that $\mathcal{L}$ and $\mathcal{L}^{-1}$ satisfy
(2.1) with $c_{\mathcal{L}}=c_{\mathcal{L}^{-1}}=C_{T}^{2}d_{1}d_{2}$, where

$\displaystyle
C_{T}=\max\left\\{\left|T_{i,j}\right|+\left|\left(T^{-1}\right)_{i,j}\right|+\left|\tilde{T}_{i^{\prime},j^{\prime}}\right|+\left|\left(\tilde{T}^{-1}\right)_{i^{\prime},j^{\prime}}\right|:i,j\in[d_{1}],i^{\prime},j^{\prime}\in[d_{2}]\right\\}<\infty.$

Note that for any $d_{1}\times d_{2}$ matrix $X$, we have

$\displaystyle\left|\left(TX\tilde{T}\right)_{i,j}\right|=\left|\sum_{k=1}^{d_{1}}\sum_{l=1}^{d_{2}}T_{i,k}X_{k,l}\tilde{T}_{l,j}\right|\leq
C_{T}^{2}\sum_{k=1}^{d_{1}}\sum_{l=1}^{d_{2}}\left|X_{k,l}\right|,\quad\forall
i\in[d_{1}],j\in[d_{2}].$

Hence, by Cauchy-Schwarz inequality, for any $d_{1}\times d_{2}$ matrices
$X,Y$,

$\displaystyle\left\|\mathcal{L}(Y)-\mathcal{L}(X)\right\|^{2}=\sum_{i=1}^{d_{1}}\sum_{j=1}^{d_{2}}\left|\left(T(X-Y)\tilde{T}\right)_{i,j}\right|^{2}\leq
C_{T}^{4}d_{1}d_{2}\left(\sum_{k=1}^{d_{1}}\sum_{l=1}^{d_{2}}\left|(X-Y)_{k,l}\right|\right)^{2}$
$\displaystyle\leq
C_{T}^{4}d_{1}^{2}d_{2}^{2}\sum_{k=1}^{d_{1}}\sum_{l=1}^{d_{2}}\left|(X-Y)_{k,l}\right|^{2}=C_{T}^{4}d_{1}^{2}d_{2}^{2}\left\|X-Y\right\|^{2}.$
(2.4)

Similarly,

$\displaystyle\left\|\mathcal{L}^{-1}(Y)-\mathcal{L}^{-1}(X)\right\|\leq
C_{T}^{2}d_{1}d_{2}\left\|X-Y\right\|.$ (2.5)

The proof is concluded by (2.1) and (2.5). ∎

###  2.2 Hitting probabilities

In this subsection, we collect some results from [Xiao2009] and [Lee2023] on
the hitting probabilities of Gaussian random fields that satisfy conditions
(A1) \- (A5).

Let $n\in\mathbb{N}$, let
$W=\\{(W_{1}(t),\ldots,W_{n}(t)):t\in\mathbb{R}_{+}^{N}\\}$ be an
$n$-dimensional Gaussian random field, whose entries are independent copies of
$\\{\xi(t):t\in\mathbb{R}_{+}^{N}\\}$. In view of Lemma 1.2, the following
lemma is a direct consequence of [Xiao2009]*Theorem 2.1, Theorem 2.3 and
Theorem 2.5.

######  Lemma 2.3.

[Xiao2009]*Theorem 2.1, Theorem 2.3 and Theorem 2.5 Consider the interval $I$
of the form (1.10). Suppose that the assumptions (A1), (A4) and (A5) hold.
Then we have the following:

  1. 1.

Let $B\subseteq\mathbb{R}^{n}$ be a Borel set, then there exist positive
constants $c_{5},c_{6}$ that depend only on $I$, $B$, $H$, such that

$\displaystyle c_{5}\mathcal{C}_{n-Q}(B)\leq\mathbb{P}\left(W^{-1}(B)\cap
I\not=\emptyset\right)\leq c_{6}{\mathbf{H}}_{n-Q}(B),$

where $Q=\sum_{j=1}^{N}\frac{1}{H_{j}}$.

  2. 2.

Let $B\subseteq\mathbb{R}^{n}$ be a Borel set such that $\dim_{{}_{\rm
H}}(B)\geq n-Q$. Then the following statements hold:

    1. (a)

Almost surely,

$\displaystyle\dim_{{}_{\rm H}}(W^{-1}(B)\cap I)\leq\min_{1\leq i\leq
N}\bigg{\\{}\sum_{j=1}^{i}\dfrac{H_{i}}{H_{j}}+N-i-
H_{i}(n-\dim_{H}(B))\bigg{\\}}.$

    2. (b)

Assume that $\dim_{{}_{\rm H}}(B)>n-Q$ and there is a finite constant
$c_{7}\geq 1$ with the following property: For every
$\eta\in(0,\,\dim_{{}_{\rm H}}(B))$ there is a finite Borel measure
$\mu_{\eta}$ with compact support in $B$ such that

$\mu_{\eta}\big{(}\mathfrak{B}_{\rho}(x)\big{)}\leq
c_{7}\,\rho^{\eta}\qquad\hbox{ for all }x\in\mathbb{R}^{n}\hbox{ and }\rho>0.$
(2.6)

Then with positive probability,

$\displaystyle\dim_{{}_{\rm H}}(W^{-1}(B)\cap I)\geq\min_{1\leq i\leq
N}\bigg{\\{}\sum_{j=1}^{i}\dfrac{H_{i}}{H_{j}}+N-i-
H_{i}(n-\dim_{H}(B))\bigg{\\}}.$

The key feature of Condition (2.6) is that the constant $c_{7}$ is independent
of $\eta$, even though the probability measure $\mu_{\eta}$ may depend on
$\eta$. For the proofs of Theorems 1.1, 1.2, 1.2 and 1.4, we take $\mu_{\eta}$
as the restriction of the Lebesgue measure on the manifolds
${\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$,
${\mathbf{H}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$,
${\mathbf{M}}_{\mathbb{R}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$,
${\mathbf{M}}_{\mathbb{C}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$ respectively.
Both of these measures are independent of $\eta$ and they satisfy (2.6).

Lemma 2.3 has less useful information when $\dim_{{}_{\rm H}}(B)=d-Q$. This
case is referred to the critical dimension of hitting probability, and is
studied in [Lee2023]. The following lemma is a case of [Lee2023]*Corollary
2.4.

######  Lemma 2.4.

[Lee2023]*Corollary 2.4 Let Assumptions (A1)-(A3) hold and let
$B\subset\mathbb{R}^{n}$ be a bounded set. Assume that $n>Q$ and
$\dim_{{}_{\rm H}}(B)=n-Q$, such that

$\displaystyle\lambda_{n}\left(B^{(r)}\right)\leq
C_{B}r^{Q}\left(\ln\ln(1/r)\right)^{\kappa}$ (2.7)

for a constant $\kappa<1$, where $\lambda_{n}$ is the $n$-dimensional Lebesgue
measure and

$\displaystyle B^{(r)}=\left\\{x\in\mathbb{R}^{n}:\inf_{y\in F}|x-y|\leq
r\right\\}$

is the $r$-neighborhood of $F$. Then $X^{-1}(F)\cap I=\emptyset$ almost
surely.

The next lemma, which is [Lee2023]*Lemma 4.1, helps to verify the condition
(2.7) of Lemma 2.4.

######  Lemma 2.5.

Let $J\subset\mathbb{R}^{m}$ be a compact interval and $G:J\to\mathbb{R}^{n}$
be a Lipschitz function, where $1\leq m\leq n$. Let $F=G(J)$, then, we have

$\displaystyle\lambda_{n}(F^{(r)})\leq Cr^{n-m}$

for all $r>0$ small, where $C$ is a positive constant only depending on
$J,m,n$ and the Lipschitz constant of the function $G$.

The following corollary says that the matrix linear transformation
$\mathcal{L}$ preserve the condition (2.7). The proof is a direct application
of (2.1) and (2.5), and is omitted.

######  Corollary 2.2.

Let $T,\tilde{T}$ and $\mathcal{L}$ be as in Lemma 2.2. If (2.7) holds for
$B$, then it also holds for $\mathcal{L}(B)$.

###  2.3 Manifolds and Lie group

In this subsection, we present some results in manifolds and Lie groups from
[Lee2013]. We start with the some conceptions in manifolds.

######  Definition 2.

Suppose $M$ and $N$ are smooth manifolds with or without boundary. A smooth
map $F:M\to N$ is called a smooth submersion if its differential is surjective
at each point. It is called a smooth immersion if its differential is
injective at each point.

The following lemma is part of [Lee2013]*Proposition 4.28, which says that a
smooth submersion is always open.

######  Lemma 2.6.

[Lee2013]*Proposition 4.28 Let $M$ and $N$ be smooth manifolds, and suppose
$\pi:M\to N$ is a smooth submersion. Then $\pi$ is an open map.

######  Definition 3.

Suppose $M$ and $N$ are smooth manifolds with or without boundary. If
$\pi:M\to N$ is any continuous map, a section of $\pi$ is a continuous right
inverse for $\pi$, i.e., a continuous map $\sigma:N\to M$ such that
$\pi\circ\sigma=\mathrm{Id}_{N}$. A local section of $\pi$ is a continuous map
$\sigma:U\to M$ defined on some open subset $U\subseteq N$ and satisfying the
analogous relation $\pi\circ\sigma=\mathrm{Id}_{U}$.

######  Lemma 2.7.

[Lee2013]*Theorem 4.26 Suppose $M$ and $N$ are smooth manifolds and $\pi:M\to
N$ is a smooth map. Then $\pi$ is a smooth submersion if and only if every
point of $M$ is in the image of a smooth local section of $\pi$.

In the following, we turn to the conceptions in Lie groups. We begin with the
definition of Lie group.

######  Definition 4.

A Lie group is a smooth manifold $G$ (without boundary) that is also a group
in the algebraic sense, such that the mapping of group multiplication and
group inverse are both smooth.

Next, we define the group action, which plays an important role in defining
quotient map.

######  Definition 5.

Let $G$ be a group and $M$ be a set.

  * •

A left action of $G$ on $M$ is a map $G\times M\to M$ with the notation
$(g,p)\mapsto g\cdot p$ such that

$\displaystyle g_{1}\cdot(g_{2}\cdot p)=(g_{1}g_{2})\cdot p\ \mathrm{and}\
e\cdot p=p,\quad\forall g_{1},g_{2}\in G,p\in M,$

where $e$ is the identity element of the group $G$.

  * •

A right action of $G$ on $M$ is a map $M\times G\to M$ with the notation
$(p,g)\mapsto p\cdot g$ such that

$\displaystyle(p\cdot g_{1})\cdot g_{2}=p\cdot(g_{1}g_{2})\ \mathrm{and}\
p\cdot e=p,\quad\forall g_{1},g_{2}\in G,p\in M.$

  * •

If in addition $M$ is a smooth manifold with or without boundary, $G$ is a Lie
group, and the defining map $G\times M\to M$ ($M\times G\to M$ resp.) is
smooth, then the left (right resp.) action is said to be a smooth action.

  * •

The action is said to be free if the only element of $G$ that fixes any
element of $M$ is the identity element.

  * •

For each $p\in M$, we denote by $G\cdot p=\\{g\cdot p:g\in G\\}$ the orbit of
$p$. The set of orbits is denoted by $M/G$, and is called the orbit space of
the action if it equips the quotient topology.

######  Remark 2.1.

The set of $d$-dimensional orthogonal matrices ${\mathbf{O}}(d)$ and the set
of $d$-dimensional unitary matrices ${\mathbf{U}}(d)$ are compact Lie groups.
One can show that ${\mathbf{O}}(d)$ is a smooth submanifold of
$\mathbb{R}^{d^{2}}$ of dimension $d(d-1)/2$ and ${\mathbf{U}}(d)$ is a smooth
submanifold of $\mathbb{C}^{d^{2}}\cong\mathbb{R}^{2d^{2}}$ of dimension
$d^{2}$. See, e.g., [Jaramillo2018]*page 7.

Now we define the proper action of Lie group.

######  Definition 6.

A continuous left action of a Lie group $G$ on a manifold $M$ is said to be a
proper left action if the map $G\times M\to M\times M$ given by
$(g,p)\mapsto(g\cdot p,p)$ is a proper map. The proper right action can be
defined similarly.

The following lemma can be used to verify that a action is proper.

######  Lemma 2.8.

[Lee2013]*Corollary 21.6 Every continuous action by a compact Lie group on a
manifold is proper.

Now we are able to introduce the following results, which is known as the
Quotient Manifold Theorem.

######  Lemma 2.9.

[Lee2013]*Theorem 21.10 Suppose $G$ is a Lie group acting smoothly, freely,
and properly on a smooth manifold $M$. Then the orbit space $M/G$ is a
topological manifold of dimension equal to $\dim M-\dim G$, and has a unique
smooth structure with the property that the quotient map $\pi:M\to M/G$ is a
smooth submersion.

##  3 Multiple collisions of eigenvalue processes

###  3.1 The real case: proof of Theorem 1.1

In this subsection, we deal with the matrix (1.11) for $\beta=1$ and prove
Theorem 1.1.

Fix $r\in\mathbb{N}$, $l_{1},\ldots,l_{r}\in\\{1,\ldots,d\\}$ satisfying
$\sum_{j=1}^{r}l_{j}\leq d$. For simplicity, we denote
$l=d-\sum_{j=1}^{r}(l_{j}-1)$ be the number of distinct eigenvalues, and
$L_{i}=d-\sum_{j=i}^{r}l_{j}+1$. Then one can check that $L_{1}=l-r+1$.

Firstly, we provide an upper bound for the dimension of
${\mathbf{S}}(d;l_{1},\ldots,l_{r})$ given in (1.18). For the dimension of
${\mathbf{S}}(d;l_{1},\ldots,l_{r})$, the upper bound
$\frac{1}{2}[d(d+1)-\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)]$ is a direct consequence
of Lemma 3.1 below.

######  Lemma 3.1.

Let $\Delta:\mathbb{R}^{l}\rightarrow{\mathbf{D}}(d)$ be a function that maps
any vector $u=(u_{1},\ldots,u_{l})\in\mathbb{R}^{l}$ to a $d\times d$ diagonal
matrix $\Delta(u)$ with entries given by

$\displaystyle\Delta_{i,i}(u)=\begin{cases}u_{i},&1\leq i\leq L_{1}-1;\\\
u_{l-r+j},&L_{j}\leq i\leq L_{j}+l_{j}-1,j=1,\ldots,r.\end{cases}$ (3.1)

Then there exists a compactly supported smooth function
$\Gamma:\mathbb{R}^{\frac{1}{2}[d(d-1)-\sum_{j=1}^{r}l_{j}(l_{j}-1)]}\rightarrow\mathbf{O}(d)$,
such that the mapping

$G:\mathbb{R}^{l}\times\mathbb{R}^{\frac{1}{2}[d(d-1)-\sum_{j=1}^{r}l_{j}(l_{j}-1)]}\rightarrow{\mathbf{S}}(d)$

given by

$\displaystyle
G(u,v)=\Gamma(v)\Delta(u)\Gamma(v)^{*},~{}~{}(u,v)\in\mathbb{R}^{l}\times\mathbb{R}^{\frac{1}{2}[d(d-1)-\sum_{j=1}^{r}l_{j}(l_{j}-1)]},$
(3.2)

satisfies

$\displaystyle{\mathbf{S}}(d;l_{1},\ldots,l_{r})\subseteq\left\\{x\in\mathbb{R}^{d(d+1)/2}:\
x\in\mathrm{Im}(G)\right\\},$ (3.3)

where $\mathrm{Im}(G)$ is the image of the mapping $G$.

###### Proof.

We denote the set of matrices

$\displaystyle{\mathbf{O}}(d;l_{1},\ldots,l_{r})=\big{\\{}B=\mathrm{diag}(b_{1},\ldots,b_{l-r},P_{1},\ldots,P_{r})\in{\mathbf{O}}(d):$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
b_{i}\in{\mathbf{O}}(1),P_{j}\in{\mathbf{O}}(l_{j}),\forall 1\leq j\leq
r,1\leq i\leq l-r\big{\\}}.$ (3.4)

With the help of Remark 2.1, one can easily check that
${\mathbf{O}}(d;l_{1},\ldots,l_{r})$ equipped with the matrix multiplication
is a Lie subgroup of ${\mathbf{O}}(d)$ with dimension
$\sum_{j=1}^{r}l_{j}(l_{j}-1)/2$. Thus, we can define the right action of the
Lie group ${\mathbf{O}}(d;l_{1},\ldots,l_{r})$ on the manifold
${\mathbf{O}}(d)$ as the matrix multiplication. Note that ${\mathbf{O}}(d)$ is
also a Lie group, which means that the group multiplication map and the group
inverse map are smooth. Thus, the right action is smooth. Moreover, by the
compactness of ${\mathbf{O}}(d)$ and Lemma 2.8, the right action is proper.
Furthermore, by checking matrix multiplication, we can see that the right
action is free. Therefore, by Lemma 2.9, the orbit space
${\mathbf{O}}(d)/{\mathbf{O}}(d;l_{1},\ldots,l_{r})$ is a smooth manifold of
dimension $\frac{1}{2}[d(d-1)-\sum_{j=1}^{r}l_{j}(l_{j}-1)]$, and the quotient
map

$\displaystyle\pi:{\mathbf{O}}(d)\longrightarrow{\mathbf{O}}(d)/{\mathbf{O}}(d;l_{1},\ldots,l_{r})$

is a smooth submersion. Furthermore, by Lemma 2.6, $\pi$ is an open map.

For $\epsilon>0$, let $I_{\epsilon}$ be the following open interval

$I_{\epsilon}=(-\epsilon,\epsilon)^{\frac{1}{2}[d(d-1)-\sum_{j=1}^{r}l_{j}(l_{j}-1)]}.$
(3.5)

For any $A\in{\mathbf{O}}(d)$, by the definition of chart (see e.g.
[Tu2011]*Definition 5.1), there exists a neighbourhood of
$\pi(A)\in{\mathbf{O}}(d)/{\mathbf{O}}(d;l_{1},\ldots,l_{r})$ that is smoothly
diffeomorphic to $I_{\epsilon}$. Moreover, since $\pi$ is a submersion, by
Lemma 2.7, there exists a smooth local section $\sigma_{\pi}$ from an open
neighbourhood of $\pi(A)$ to ${\mathbf{O}}(d)$, such that $\sigma_{\pi}$ maps
$\pi(A)$ to $A$ and $\pi\circ\sigma_{\pi}$ is identity. Note that $\pi$ is
open. Thus, we can choose $\delta$ small enough, such that there exist a
positive number $\epsilon$ (which may depend on $A$), an open neighbourhood
$U(\pi(A))$ of $\pi(A)$ in the domain of $\sigma_{\pi}$ and a smooth
diffeomorphism

$\displaystyle\phi:I_{\epsilon}\rightarrow
U(\pi(A))\subset\left({\mathbf{O}}(d)/{\mathbf{O}}(d;l_{1},\ldots,l_{r})\right)\cap\mathfrak{\pi}(\mathfrak{B}_{\delta}(A))$
(3.6)

satisfying $\phi(0)=\pi(A)$. We write $U_{\phi}(\pi(A))$ for $U(\pi(A))$ in
the following to emphasize that the neighbourhood is associated with $\phi$.
Here $\mathfrak{B}_{\delta}(A)$ is the open ball with radius $\delta$ centered
at $A$ in the space $\mathbf{O}(d)$ under the Frobenius norm $\|\cdot\|$ given
by (1.22).

Therefore, we can construct a smooth function

$\displaystyle\Gamma:\mathbb{R}^{\frac{1}{2}[d(d-1)-\sum_{j=1}^{r}l_{j}(l_{j}-1)]}\rightarrow{\mathbf{O}}(d)$

supported on $I_{2\epsilon}$, such that

$\displaystyle\Gamma(v)=\sigma_{\pi}\circ\phi(v),\forall v\in I_{\epsilon}.$
(3.7)

Since $\phi$ is a diffeomorphism, the set
$U_{\phi}(\pi(A))=\phi(I_{\epsilon})$ is a subset of
${\mathbf{O}}(d)/{\mathbf{O}}(d;l_{1},\ldots,l_{r})$ that is open and contains
$\pi(A)$. Hence, the collection of the sets
$\\{U_{\phi}(\pi(A)):\pi(A)\in{\mathbf{O}}(d)/{\mathbf{O}}(d;l_{1},\ldots,l_{r})\\}$
is an open cover for ${\mathbf{O}}(d)/{\mathbf{O}}(d;l_{1},\ldots,l_{r})$. By
the compactness of ${\mathbf{O}}(d)$ and the continuity of $\pi$,
${\mathbf{O}}(d)/{\mathbf{O}}(d;l_{1},\ldots,l_{r})$ is compact. Then one can
find a finite subcover $\\{U_{\phi^{(i)}}(\pi(A_{i})),i=1,\dots,M\\}$ for some
$M\in\mathbb{N}$, such that

$\displaystyle{\mathbf{O}}(d)/{\mathbf{O}}(d;l_{1},\ldots,l_{r})=\bigcup_{i=1}^{M}U_{\phi^{(i)}}(\pi(A_{i})),$
(3.8)

where $A_{1},\dots,A_{M}$ are distinct matrices in $\mathbf{O}(d)$,
$\phi^{(1)},\ldots,\phi^{(M)}$ are smooth mappings of the form (3.6), and
$U_{\phi^{(i)}}(\pi(A_{i}))=\phi^{(i)}(I_{\epsilon_{i}})$.

For $1\leq i\leq M$, we define the mapping $G^{(i)}:\mathbb{R}^{l}\times
I_{\epsilon_{i}}\rightarrow{\mathbf{S}}(d)$ by

$\displaystyle G^{(i)}(u,v)=\Gamma^{(i)}(v)\Delta(u)\Gamma^{(i)}(v)^{*},$
(3.9)

for $v\in I_{\epsilon_{i}}$ and $u\in\mathbb{R}^{l}$.

For an arbitrary fixed $x\in{\mathbf{S}}(d;l_{1},\ldots,l_{r})$, we have the
spectral decomposition $x=QDQ^{*}$ for some $Q\in{\mathbf{O}}(d)$ and
$D\in{\mathbf{D}}(d)$. By rearranging the diagonal entries of $D$ and the
columns of $Q$, we can assume that
$D_{L_{j},L_{j}}=\cdots=D_{L_{j}+l_{j}-1,L_{j}+l_{j}-1}$ for $1\leq j\leq r$.
By (3.8), there exists $i_{0}\in\\{1,\dots,M\\}$ such that $\pi(Q)\in
U_{\phi^{(i_{0})}}(\pi(A_{i_{0}}))$, and hence one can find $v\in
I_{\epsilon_{i_{0}}}$ such that $\pi(Q)=\phi^{(i_{0})}(v)$. By the
relationship (3.7), we can write
$\Gamma^{(i_{0})}(v)=\sigma_{\pi}\circ\pi(Q)$. Note that
$\sigma_{\pi}\circ\pi(Q)$ and $Q$ are in the same orbit space of $\pi$, we
have $\sigma_{\pi}\circ\pi(Q)=Q\cdot g$, where $g$ is an element in the Lie
group ${\mathbf{O}}(d;l_{1},\ldots,l_{r})$. Let
$u=(D_{1,1},D_{2,2},\ldots,D_{L_{1}-1,L_{1}-1},D_{L_{1},L_{1}},D_{L_{2},L_{2}},\ldots,D_{L_{r},L_{r}})\in\mathbb{R}^{l}$
then $D=\Delta(u)$. Thus, by the relationship (3.7), we have

$\displaystyle x=QDQ^{*}=Q(gDg^{*})Q^{*}=(Q\cdot g)D(Q\cdot
g)^{*}=\Gamma^{(i_{0})}(v)\Delta(u)\Gamma^{(i_{0})}(v)^{*}=G^{(i_{0})}(u,v).$

Since $x\in{\mathbf{S}}(d;l_{1},\ldots,l_{r})$ is arbitrarily chosen, we
conclude that

$\displaystyle{\mathbf{S}}(d;l_{1},\ldots,l_{r})\subseteq\left\\{x\in\mathbb{R}^{d(d+1)/2}:x\in\bigcup_{i=1}^{M}\mathrm{Im}(G^{(i)})\right\\}.$

Finally, let $\epsilon=\max\\{\epsilon_{1},\ldots,\epsilon_{M}\\}$, then for
any smooth function $\Gamma$ supported on $I_{4M\epsilon}$ satisfying

$\displaystyle\Gamma(y)=\Gamma^{(j+1)}\left(y-(4j\epsilon,0,\dots,0)\right),$

for $y\in
I_{2\epsilon}+\left\\{\left(4j\epsilon,0,\ldots,0\right)\right\\}\subseteq\mathbb{R}^{\frac{1}{2}[d(d-1)-\sum_{j=1}^{r}l_{j}(l_{j}-1)]}$
for some $j\in\\{0,1,\dots,M-1\\}$, the mapping $G$ defined by (3.2) satisfies
(3.3). ∎

######  Remark 3.1.

We would like to remark that Lemma 3.1 was obtained in
[Jaramillo2018]*Proposition 4.5 for the case $r=1,l_{1}=2$ and in [Xiao2021]
for the case $r=1$. In the two references, the mapping $\Gamma$ is constructed
firstly by finding all the eigenvectors associated to single eigenvalues, and
then by finding eigenvectors associated with multiple eigenvalues via
orthogonal extension and Gram-Schmidt orthonormalization. However, the
argument in [Jaramillo2018, Xiao2021] can not be extended to the case $r\geq
2$, since the orthogonal complement of the space spanned by eigenvectors
associated to single eigenvalues is a direct sum of eigenspaces of multiple
eigenvalues.

The following lemma is about the continuity of the spectral decomposition on
${\mathbf{S}}(d;l_{1},\ldots,l_{r})$, and it is an extension of
[Jaramillo2018]*Lemma 4.3.

######  Lemma 3.2.

Let $A\in{\mathbf{S}}(d;l_{1},\ldots,l_{r})$ be a symmetric matrix with
spectral decomposition $A=PDP^{*}$ for some $P\in{\mathbf{O}}(d)$ and
$D\in{\mathbf{D}}(d)$, such that $|\mathbf{Spec}(A)|=l$. Then for any
$\epsilon>0$, there exists $\delta>0$, such that for all
$B\in{\mathbf{S}}(d;l_{1},\ldots,l_{r})$ satisfying

$\displaystyle\max_{1\leq i,j\leq d}|A_{i,j}-B_{i,j}|<\delta,$

we have $|\mathbf{Spec}(B)|=l$, and there exists a spectral decomposition
$B=QFQ^{*}$, where $Q\in{\mathbf{O}}(d)$ and $F\in{\mathbf{D}}(d)$ satisfy

$\displaystyle\max_{1\leq i\leq d}|D_{i,i}-F_{i,i}|<\epsilon,\quad\max_{1\leq
i,j\leq d}|Q_{i,j}-P_{i,j}|<\epsilon.$ (3.10)

###### Proof.

We follow the idea of the proof of [Jaramillo2018]*Lemma 4.3. The first
inequality of (3.10), which describes the continuity of the eigenvalues,
follows directly from the continuity of the functions $E_{1},\ldots,E_{d}$
which are introduced in Section 1.4. The second inequality of (3.10) claims
that eigenvectors, considered as functions of matrices in
${\mathbf{S}}(d;l_{1},\ldots,l_{r})$, are continuous at
$A\in{\mathbf{S}}(d;l_{1},\ldots,l_{r})$ with $\mathbf{Spec}(A)=l$. The key
idea to prove this is to represent eigenprojections as matrix-valued Cauchy
integrals.

Noting that $A\in{\mathbf{S}}(d;l_{1},\ldots,l_{r})$ and
$|\mathbf{Spec}(A)|=l$, without loss of generality we assume

$\displaystyle
D_{1,1}<\cdots<D_{L_{1},L_{1}}=\cdots=D_{L_{2}-1,L_{2}-1}<D_{L_{2},L_{2}}=\cdots=D_{L_{3}-1,L_{3}-1}<\cdots<D_{L_{r},L_{r}}=\cdots=D_{d,d}.$
(3.11)

Denote by $\tilde{I}=[L_{1}-1]\cup\\{L_{j}:1\leq j\leq r\\}$ the indexes of
distinct eigenvalues. For $i\in\tilde{I}$, let
$\mathcal{C}_{i}^{A}\subseteq\mathbb{C}\setminus\mathbf{Spec}(A)$ be any
smooth closed curve around $D_{i,i}$ and denote by $\mathcal{I}_{i}^{A}$ the
closure of the interior of $\mathcal{C}_{i}^{A}$. We can choose the curves
$\\{\mathcal{C}_{i}^{A}:i\in\tilde{I}\\}$ with sufficiently small diameters so
that $\\{\mathcal{I}_{i}^{A}:i\in\tilde{I}\\}$ are disjoint. For simplicity,
let $\mathcal{C}_{i}^{A}=\mathcal{C}_{L_{j}}^{A}$ and
$\mathcal{I}_{i}^{A}=\mathcal{I}_{L_{j}}^{A}$ for $L_{j}<i\leq L_{j}+l_{j}-1$,
$1\leq j\leq r$.

By the continuity of the functions
$E_{1},E_{2},\ldots,E_{L_{1}},E_{L_{2}},\ldots,E_{L_{r}}$ and (3.11), there
exists $\delta>0$, such that for all $B$ in the $\delta$-neighbourhood in
${\mathbf{S}}(d;l_{1},\ldots,l_{r})$ of $A$

$\displaystyle
U_{\delta}=\left\\{B\in{\mathbf{S}}(d;l_{1},\ldots,l_{r}):\max_{1\leq i,j\leq
d}|A_{i,j}-B_{i,j}|<\delta\right\\},$

we have $E_{1}(B)<\cdots<E_{L_{1}}(B)<E_{L_{2}(B)}<\ldots<E_{L_{r}}(B)$ and
$E_{i}(B)\in\mathcal{I}_{i}^{A}$ for $i\in\tilde{I}$. Noting that
$U_{\delta}\subseteq{\mathbf{S}}(d;l_{1},\ldots,l_{r})$, for all $B\in
U_{\delta}$, we have

$\displaystyle
E_{1}(B)<\cdots<E_{L_{1}}(B)=\cdots=E_{L_{2}-1}(B)<E_{L_{2}}(B)=\cdots=E_{L_{3}-1}(B)<\ldots<E_{L_{r}}(B)=\cdots=E_{d}(B),$
(3.12)

and

$\displaystyle E_{i}(B)\in\mathcal{I}_{i}^{A},\quad 1\leq i\leq d.$ (3.13)

Thus, for any $i\in[d]$, for any $\zeta\in\mathcal{C}_{i}^{A}$, (3.13) implies
that the matrix $\zeta I_{d}-B$ is invertible. Hence, we can define the
mappings $\Theta_{i}^{A}:U_{\delta}\rightarrow{\mathbf{S}}(d)$ by the
following matrix-valued Cauchy integrals:

$\displaystyle\Theta_{i}^{A}(B)=\dfrac{1}{2\pi\iota}\oint_{\mathcal{C}_{i}^{A}}(\zeta
I_{d}-B)^{-1}d\zeta\,.$

For $B_{1},B_{2}\in U_{\delta}$, we have

$\displaystyle\left(\zeta I_{d}-B_{1}\right)^{-1}-\left(\zeta
I_{d}-B_{2}\right)^{-1}=\left(\zeta
I_{d}-B_{1}\right)^{-1}\left(B_{1}-B_{2}\right)\left(\zeta
I_{d}-B_{2}\right)^{-1},$

which implies that the mappings $\Theta_{i}^{A}$ are continuous with respect
to $B$ for $B\in U_{\delta}$. By [Lax2002]*page 200, Theorem 6, the matrix
$\Theta_{i}^{A}(B)$ is a projection over the sum of the eigenspaces associated
with eigenvalues of $B$ that are inside $\mathcal{I}_{i}^{A}$. Hence, by
(3.12) and (3.13), $\Theta_{i}^{A}(B)$ is a projection over the eigenspace
$\mathcal{E}_{E_{i}(B)}^{B}$ for $1\leq i\leq d$, noting that
$\\{\mathcal{I}_{i}^{A}:i\in\tilde{I}\\}$ are disjoint.

For $1\leq j\leq L_{1}-1$, we define

$\displaystyle
w^{j}=\dfrac{\Theta_{j}^{A}(B)P_{*,j}}{\left\|\Theta_{j}^{A}(B)P_{*,j}\right\|},$
(3.14)

which are unit eigenvectors of $E_{j}(B)$ for $j=1,\dots,L_{1}-1$. Recall that
$\Theta_{j}^{A}(B)$ is a continuous function of $B$ for $B\in U_{\delta}$ with
$\delta$ sufficiently small and that $\Theta_{j}^{A}(A)P_{*,j}=P_{*,j}$, for
all $1\leq j\leq d$. Hence, for all $1\leq i\leq r$, for $L_{i}\leq j\leq
L_{i}+l_{i}-1$, noting that
$\mathcal{E}_{E_{L_{i}}(B)}^{B}=\cdots=\mathcal{E}_{E_{L_{i}+l_{i}-1}(B)}^{B}$
is a $l_{i}$-dimensional vector space and the vectors
$P_{*,L_{i}},\ldots,P_{*,L_{i}+l_{i}-1}$ are orthogonal, we can see that
$\Big{\\{}\Theta_{L_{i}}^{A}(B)P_{*,L_{i}},\cdots,\Theta_{L_{i}+l_{i}-1}^{A}(B)P_{*,L_{i}+l_{i}-1}\Big{\\}}$
is the set of linearly independent eigenvectors associated with the same
eigenvalues $E_{L_{i}}$ for $\delta$ being suggiciently small. Thus, we define
iteratively for $B\in U_{\delta}$ with sufficiently small $\delta$, by
applying the Gram-Schmidt orthonormalizing process to obtain the following
unit eigenvectors of $E_{L_{i}}(B)$

$\begin{cases}w^{L_{i}}(B)=\dfrac{\Theta_{L_{i}}^{A}(B)P_{*,L_{i}}}{\left\|\Theta_{L_{i}}^{A}(B)P_{*,L_{i}}\right\|},\\\
w^{j}(B)=\dfrac{\dfrac{\Theta_{j}^{A}(B)P_{*,j}}{\big{\|}\Theta_{j}^{A}(B)P_{*,j}\big{\|}}-\sum_{j^{\prime}=L_{i}}^{j-1}\Big{\langle}\dfrac{\Theta_{j}^{A}(B)P_{*,j}}{\|\Theta_{j}^{A}(B)P_{*,j}\|},w^{j^{\prime}}(B)\Big{\rangle}w^{j^{\prime}}(B)}{\bigg{\|}\dfrac{\Theta_{j}^{A}(B)P_{*,j}}{\|\Theta_{j}^{A}(B)P_{*,j}\|}-\sum_{j^{\prime}=L_{i}}^{j-1}\Big{\langle}\dfrac{\Theta_{j}^{A}(B)P_{*,j}}{\|\Theta_{j}^{A}(B)P_{*,j}\|},w^{j^{\prime}}(B)\Big{\rangle}w^{j^{\prime}}(B)\bigg{\|}},\quad
L_{i}+1\leq j\leq L_{i}+l_{i}-1.\end{cases}$ (3.15)

Also note that the inner product $\langle u,v\rangle$ is a continuous function
of $(u,v)$, and hence $\omega^{j}(B),j=1,\dots,d$, defined by (3.14) and
(3.15) are continuous functions of $B$ in a sufficiently small neighborhood
$U_{\delta}$ of $A$ with $\omega^{j}(A)=P_{*,j}$. Thus, for any $\epsilon>0$,
one can find a sufficiently small positive constant $\delta$, such that for
all $B\in U_{\delta}$,

$\displaystyle\max_{1\leq j\leq d}\|P_{*,j}-w^{j}(B)\|<\epsilon.$

Thus if we denote the matrix $Q=(\omega^{1}(B),\dots,\omega^{d}(B))$, then
$B=QFQ^{*}$ and the second inequality of (3.10) is satisfied. The proof is
concluded. ∎

The following result is an extension of [Jaramillo2018]*Proposition 4.7, and
it shows that the optimal upper bound for the dimension of
${\mathbf{S}}(d;l_{1},\ldots,l_{r})$ is
$\frac{1}{2}[d(d+1)-\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)]$.

######  Lemma 3.3.

For $x_{0}\in{\mathbf{S}}(d;l_{1},\ldots,l_{r})$ with
$|\mathbf{Spec}(x_{0})|=l$, there exists $\delta_{0}>0$ such that
${\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$ is a
$(\frac{1}{2}[d(d+1)-\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)])$-dimensional manifold.
In particular,
${\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$ has
positive $(\frac{1}{2}[d(d+1)-\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)])$-dimensional
Lebesgue measure.

###### Proof.

By Lemma 3.1, there exist $u_{0}\in\mathbb{R}^{l}$,
$v_{0}\in\mathbb{R}^{\frac{1}{2}[d(d-1)-\sum_{j=1}^{r}l_{j}(l_{j}-1)]}$, such
that $x_{0}=G(u_{0},v_{0})$, where the function $G$ is given by (3.2).
Moreover, without loss of generality, we may assume that
$x_{0}=G^{(1)}(u_{0},v_{0})=\Gamma^{(1)}(v_{0})\Delta(u_{0})\Gamma^{(1)}(v_{0})^{*}$,
$v_{0}\in I_{\epsilon_{1}}$ and the components of $u_{0}$ are increasing. To
avoid confusing, we rewrite (3.7) as

$\displaystyle\Gamma^{(1)}(v)=\sigma_{\pi}^{(1)}\circ\phi^{(1)}(v),\forall
v\in I_{\epsilon_{1}}.$ (3.16)

To show that the manifold
${\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$ has
dimension $\frac{1}{2}[d(d+1)-\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)]$ and has
positive $(\frac{1}{2}[d(d+1)-\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)])$-dimensional
Lebesgue measure, it is sufficient to show that there exist open sets
$U\subseteq\mathbb{R}^{l}\times I_{\epsilon_{1}}$ and
$V\subseteq{\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$,
such that the map $G^{(1)}|_{U}:U\rightarrow V$ is a homeomorphism.

Let

$\displaystyle
r_{0}=\dfrac{1}{2}\min_{\begin{subarray}{c}\mu,\lambda\in\mathbf{Spec}(x_{0})\\\
\mu\not=\lambda\end{subarray}}|\mu-\lambda|.$ (3.17)

For $\gamma_{0}\in(0,r_{0})$ that is small enough, by Lemma 3.2, there exists
$\delta_{0}$ such that for each
$x\in{\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$,
it has the spectral decomposition $x=QEQ^{*}$ with
$Q\in{\mathbf{O}}(d)\cap\mathfrak{B}_{\gamma_{0}}\left(\Gamma^{(1)}(v_{0})\right)$
and $E\in{\mathbf{D}}(d)\cap\mathfrak{B}_{\gamma_{0}}(\Delta(u_{0}))$, such
that $|\mathbf{Spec}(x)|=l$. Then by the definition (3.17) of $r_{0}$, we have

$\displaystyle
E_{1,1}<E_{2,2}<\cdots<E_{L_{1}-1,L_{1}-1}<E_{L_{1},L_{1}}=\cdots=E_{L_{2}-1,L_{2}-1}<E_{L_{2},L_{2}}=\cdots<E_{L_{r},L_{r}}=\cdots=E_{d,d}.$

Denote
$u=(E_{1,1},\dots,E_{L_{1}-1,L_{1}-1},E_{L_{1},L_{1}},E_{L_{2},L_{2}},\ldots,E_{L_{r},L_{r}})\in\mathbb{R}^{l}$,
then $E=\Delta(u)$. Recall that $\pi$ is continuous and $\phi^{(1)}$ is a
diffeomorphism. We choose $\gamma_{0}$ to be sufficiently small, then for
$Q\in{\mathbf{O}}(d)\cap\mathfrak{B}_{\gamma_{0}}\left(\Gamma^{(1)}(v_{0})\right)$,
we have $\pi(Q)\in\phi^{(1)}(I_{(\epsilon_{1}+\|v_{0}\|)/2})$, noting that
$\pi\left(\Gamma^{(1)}(v_{0})\right)=\phi^{(1)}(v_{0})$. Thus, we choose
$v=\left(\phi^{(1)}\right)^{-1}(\pi(Q))\in I_{(\epsilon_{1}+\|v_{0}\|)/2}$,
then $\Gamma^{(1)}(v)=\sigma_{\pi}^{(1)}\left(\pi(Q)\right)$ is an orthogonal
matrix in the same orbit space as $Q$. Hence, there exists
$g\in{\mathbf{O}}(d;l_{1},\ldots,l_{r})$ such that
$\sigma_{\pi}^{(1)}\circ\pi(Q)=Q\cdot g$. Therefore,

$\displaystyle x=QEQ^{*}=Q(gEg^{*})Q^{*}=(Q\cdot g)E(Q\cdot
g)^{*}=\Gamma^{(1)}(v)\Delta(u)\Gamma^{(1)}(v)^{*}=G^{(1)}(u,v).$ (3.18)

Now we choose
$V={\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$,
then by (3.18), we have $V\subset G^{(1)}(\mathbb{R}^{l}\times
I_{(\epsilon_{1}+\|v_{0}\|)/2})$. We choose $U=(G^{(1)})^{-1}(V)$, then
$\left.G^{(1)}\right|_{U}$ is surjective. The continuity of the mapping
$G^{(1)}$ implies that $U$ is open in $\mathbb{R}^{l}\times
I_{(\epsilon_{1}+\|v_{0}\|)/2}$. To show $\left.G^{(1)}\right|_{U}$ is a
homeomorphism, it suffices to show the following conditions are satisfied:

  1. (a1)

$\left.G^{(1)}\right|_{U}$ is injective;

  2. (a2)

$\left(\left.G^{(1)}\right|_{U}\right)^{-1}$ is continuous over $V$.

Firstly, we show that
$\left.G^{(1)}\right|_{U}:U\to{\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$
is injective. Suppose that for
$x\in{\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$,
it has the following spectral decompositions,

$\displaystyle
x=\Gamma^{(1)}(v)\Delta(u)\Gamma^{(1)}(v)^{*}=\Gamma^{(1)}(v^{\prime})\Delta(u^{\prime})\Gamma^{(1)}(v^{\prime})^{*}.$

We aim to show that $(u,v)=(u^{\prime},v^{\prime}).$

Denote
$u^{\prime}=(u^{\prime}_{1},\ldots,u^{\prime}_{L_{1}},\ldots,u_{L_{r}}^{\prime})$.
Note that the set of eigenvalues is the set of zeros of the characteristic
polynomial, which is uniquely determined by $x$. Thus, if $u_{i}\neq
u^{\prime}_{i}$ for some $i\in\tilde{I}$, then the corresponding unit
eigenvectors $\Gamma^{(1)}_{*,i}(v)$ and $\Gamma^{(1)}_{*,i}(v^{\prime})$
belong to different eigenspaces and hence are orthogonal. On the other hand,
by (3.6), there exist $A_{1},A_{2}$ in the open ball with radius
$\delta^{(1)}$ in the space $\mathbf{O}(d)$, such that

$\displaystyle\pi\left(\Gamma^{(1)}(v)\right)=\phi^{(1)}(v)=\pi(A_{1}),\
\pi\left(\Gamma^{(1)}(v^{\prime})\right)=\phi^{(1)}(v^{\prime})=\pi(A_{2}).$

Thus, $\Gamma^{(1)}(v)$ and $\Gamma^{(1)}(v^{\prime})$ are in the same orbit
space as $A_{1}$ and $A_{2}$ respectively. Hence, there exist
$g_{1},g_{2}\in{\mathbf{O}}(d;l_{1},\ldots,l_{r})$ such that
$\Gamma^{(1)}(v)\cdot g_{1}=A_{1}$ and $\Gamma^{(1)}(v^{\prime})\cdot
g_{2}=A_{2}$. Note that $\Gamma^{(1)}(v)\cdot g_{1}$ is a matrix obtained from
$\Gamma^{(1)}(v)$ by applying orthogonal transformation to the set of
eigenvectors associate to the same eigenvalue. So is
$\Gamma^{(1)}(v^{\prime})\cdot g_{2}$. Since the $i$th columns of
$\Gamma^{(1)}(v)$ and $\Gamma^{(1)}(v^{\prime})$ belong to different
eigenspaces, we can see that the $i$th columns of $A_{1}$ and $A_{2}$ also
belong to different eigenspaces. Hence,
$\|A_{1}-A_{2}\|\geq\|(A_{1})_{*,i}-(A_{2})_{*,i}\|\geq\sqrt{2}$, which is a
contradiction if we choose $\delta^{(1)}<\sqrt{2}/2$. Therefore, we have
$u=u^{\prime}$.

Note that the components of $u$ are distinct, and the $i$th columns of
$\Gamma^{(1)}(v)$ and $\Gamma^{(1)}(v^{\prime})$ are the eigenvectors
associated to the same eigenvalue for all $1\leq i\leq d$. Thus, by the
definition of $\pi$, we have
$\pi\left(\Gamma^{(1)}(v)\right)=\pi\left(\Gamma^{(1)}(v^{\prime})\right)$. By
(3.16), we have $\phi^{(1)}(v)=\phi^{(1)}(v^{\prime})$. Therefore,
$v=v^{\prime}$ since $\phi^{(1)}$ is a diffeomorphism.

Now we show that the condition (a2) is satisfied. Consider any sequence
$\\{x_{n}\\}_{n\in\mathbb{N}}\subseteq V$, such that
$\lim_{n\rightarrow\infty}x_{n}=x\in V$. Let
$(u_{n},v_{n})=\left(\left.G^{(1)}\right|_{U}\right)^{-1}(x_{n})\in\mathbb{R}^{l}\times
I_{(\epsilon_{1}+\|v_{0}\|)/2}$ and
$(u,v)=\left(\left.G^{(1)}\right|_{U}\right)^{-1}(x)\in\mathbb{R}^{l}\times
I_{(\epsilon_{1}+\|v_{0}\|)/2}$, then

$\displaystyle
x_{n}=\Gamma^{(1)}(v_{n})\Delta(u_{n})\Gamma^{(1)}(v_{n})^{*},\quad
x=\Gamma^{(1)}(v)\Delta(u)\Gamma^{(1)}(v)^{*}.$

Note that the components of $u_{0}$ are distinct. By Lemma 3.2 and the
continuity of the functions $E_{1},\ldots,E_{d}$, it is clear that

$\displaystyle\lim_{n\rightarrow\infty}u_{n}=u.$ (3.19)

Note that for all $n\in\mathbb{N}$,
$v_{n}\in\overline{I_{(\epsilon_{1}+\|v_{0}\|)/2}}$, which is compact. Let
$v^{\prime}\in\overline{I_{(\epsilon_{1}+\|v_{0}\|)/2}}\subset
I_{\epsilon_{1}}$ be a limit point of the sequence
$\\{v_{n}\\}_{n\in\mathbb{N}}$, then there exists a subsequence
$\\{v_{m_{n}}\\}_{n\in\mathbb{N}}$, such that
$\lim_{n\rightarrow\infty}v_{m_{n}}=v^{\prime}$. By the continuity of the
mappings $\Gamma^{(1)}$ and $\Delta$, we have

$\displaystyle
x=\lim_{n\rightarrow\infty}x_{m_{n}}=\lim_{n\rightarrow\infty}\Gamma^{(1)}(v_{m_{n}})\Delta(u_{m_{n}})\Gamma^{(1)}(v_{m_{n}})^{*}=\Gamma^{(1)}(v^{\prime})\Delta(u)\Gamma^{(1)}(v^{\prime})^{*}.$

Thus, the columns of $\Gamma^{(1)}(v)$ and $\Gamma^{(1)}(v^{\prime})$ are the
eigenvectors of the same eigenvalues, which implies that
$\pi\left(\Gamma^{(1)}(v)\right)=\pi\left(\Gamma^{(1)}(v^{\prime})\right)$.
Since $\pi\circ\sigma_{\pi}^{(1)}$ is identity and $\phi^{(1)}$ is
diffeomorphism on $I_{\epsilon_{1}}$, we have $v=v^{\prime}$, which together
with the compactness of $\overline{I_{(\epsilon_{1}+\|v_{0}\|)/2}}$ implies
that

$\lim_{n\to\infty}v_{n}=v,$ (3.20)

since any limit point of the sequence $\\{v_{n}\\}_{n\in\mathbb{N}}$ has to be
$v$.

By (3.19) and (3.20), we obtain the continuity of the map
$\left(G^{(1)}\right)^{-1}$ and (a2) is proved. The proof is concluded. ∎

Now we are ready to prove Theorem 1.1.

###### Proof of Theorem 1.1.

We first prove Part (i), recalling that $\beta=1$. By Lemma 3.1, there exists
a smooth map

$\displaystyle
G:\mathbb{R}^{\frac{1}{2}[d(d+1)-\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)]}\rightarrow{\mathbf{S}}(d)$

such that ${\mathbf{S}}(d;l_{1},\ldots,l_{r})\subseteq\mathrm{Im}(G)$.
Moreover, by Lemma 3.3,

$\displaystyle\dim_{{}_{\rm
H}}\left(\mathrm{Im}(G)\right)=\dfrac{1}{2}\left(d(d+1)-\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)\right).$
(3.21)

Besides, we have

$\begin{split}&\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)\\\
&=\mathbb{P}\left(Y^{\beta}(t)\in{\mathbf{S}}(d;l_{1},\ldots,l_{r})\mathrm{\
for\ some\ }t\in I\right)\\\ &=\mathbb{P}\left(X^{\beta}(t)\in
T_{\beta}^{-1}\left({\mathbf{S}}(d;l_{1},\ldots,l_{r})-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\mathrm{\
for\ some\ }t\in I\right)\\\ &\leq\mathbb{P}\left(X^{\beta}(t)\in
T_{\beta}^{-1}\left(\mathrm{Im}(G)-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\mathrm{\
for\ some\ }t\in I\right)\\\ &=\mathbb{P}\left(X^{\beta}(I)\cap
T_{\beta}^{-1}\left(\mathrm{Im}(G)-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\not=\emptyset\right).\end{split}$
(3.22)

On one hand, in the case that

$\displaystyle\sum_{j=1}^{N}\frac{1}{H_{j}}<\dfrac{1}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2),$
(3.23)

we apply Lemma 2.3 to the last term in (3.22) to obtain

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)$
$\displaystyle\leq
c_{6}{\mathbf{H}}_{\frac{d(d+1)}{2}-\sum_{j=1}^{N}\frac{1}{H_{j}}}\left(T_{\beta}^{-1}\left(\mathrm{Im}(G)-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\right)=c_{6}{\mathbf{H}}_{\frac{d(d+1)}{2}-\sum_{j=1}^{N}\frac{1}{H_{j}}}\left(\mathrm{Im}(G)-A^{\beta}\right)$
$\displaystyle=c_{6}{\mathbf{H}}_{\frac{d(d+1)}{2}-\sum_{j=1}^{N}\frac{1}{H_{j}}}\left(\mathrm{Im}(G)\right)=0.$

In the above, the first equality follows from Corollary 2.1 and Lemma 2.2, the
second equality follows from the translation invariance of Hausdorff measure,
and the third equality follows from the inequality $\dim_{{}_{\rm
H}}(\mathrm{Im}(G))<\frac{d(d+1)}{2}-\sum_{j=1}^{N}\frac{1}{H_{j}}$ that is a
combination of (3.21) and (3.23).

On the other hand, for $K\geq 1$ large, we choose the interval

$\displaystyle F_{K}=[-K,K]^{l}\times
F^{\prime}\subseteq\mathbb{R}^{\frac{1}{2}[d(d+1)-\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)]},$

where $F^{\prime}$ is a compact interval containing the support of $\Gamma$
defined in Lemma 3.1. Then $F_{K}$ is a compact interval. We write
$G|_{F_{K}}$ as a smooth restriction of $G$ on the neighbourhood of $F_{K}$ in
the sense that $G|_{F_{K}}=G$ on $F_{K}$, and $G|_{F_{K}}=0$ on
$F_{K+1}^{\complement}$. If

$\displaystyle\sum_{j=1}^{N}\frac{1}{H_{j}}=\dfrac{1}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2),$

then Lemma 2.5 yields

$\displaystyle\lambda_{\frac{d(d+1)}{2}}\left(\left(\mathrm{Im}(G|_{F_{K}})-A^{\beta}\right)^{(r)}\right)\leq
Cr^{\frac{1}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)}=Cr^{\sum_{j=1}^{N}\frac{1}{H_{j}}}$

for all $r>0$ small, and $C$ is a constant only depending on
$G,d,H_{1},\ldots,H_{N}$. By the definition (3.2) and a similar argument as
(3.22), for $K$ large enough, we have

$\displaystyle\mathbb{P}\bigg{(}\left\\{\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right\\}$
$\displaystyle\quad\quad\quad\quad\quad\quad\cap\\{\mathbf{Spec}(Y^{\beta})\subseteq[-K,K]\\}\bigg{)}$
$\displaystyle\leq\mathbb{P}\left(\left\\{X^{\beta}(t)\in
T_{\beta}^{-1}\left(\mathrm{Im}(G)-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\mathrm{\
for\ some\ }t\in
I\right\\}\cap\left\\{\mathbf{Spec}\left(Y^{\beta}\right)\subseteq[-K,K]\right\\}\right)$
$\displaystyle\leq\mathbb{P}\left(X^{\beta}(I)\cap
T_{\beta}^{-1}\left(\mathrm{Im}(G|_{F_{K}})-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\not=\emptyset\right)$

Hence, by Lemma 2.4 and Corollary 2.2, we obtain

$\displaystyle\mathbb{P}\bigg{(}\left\\{\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right\\}$
$\displaystyle\quad\quad\quad\quad\quad\quad\cap\\{\mathbf{Spec}(Y^{\beta})\subseteq[-K,K]\\}\bigg{)}=0.$

Letting $K\to\infty$, we have

$\displaystyle\mathbb{P}\bigg{(}\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\bigg{)}=0.$

The proof of Part (i) of Theorem 1.1 is concluded.

Next, we prove Part (ii). We choose
$x_{0}\in{\mathbf{S}}(d;l_{1},\ldots,l_{r})$ satisfying
$\mathbf{Spec}(x_{0})=l$. By Lemma 3.3, there exists $\delta_{0}>0$, such that
${\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$ is a
$(\frac{1}{2}[d(d+1)-\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)])$-dimensional manifold.
Then, similarly to (3.22), by using lower bound on hitting probability in
Lemma 2.3, we have

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)$
$\displaystyle=\mathbb{P}\left(X^{\beta}(t)\in
T_{\beta}^{-1}\left({\mathbf{S}}(d;l_{1},\ldots,l_{r})-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\mathrm{\
for\ some\ }t\in I\right)$ $\displaystyle\geq\mathbb{P}\left(X^{\beta}(t)\in
T_{\beta}^{-1}\left({\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\mathrm{\
for\ some\ }t\in I\right)$ $\displaystyle=\mathbb{P}\left(X^{\beta}(I)\cap
T_{\beta}^{-1}\left({\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\not=\emptyset\right)$
$\displaystyle\geq
c_{5}\mathcal{C}_{\frac{d(d+1)}{2}-\sum_{j=1}^{N}\frac{1}{H_{j}}}\left(T_{\beta}^{-1}\left({\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\right)>0.$

In the above, the last step follows from

$\displaystyle\dim_{{}_{\rm
H}}\left(T_{\beta}^{-1}\left({\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\right)$
$\displaystyle=\dim_{{}_{\rm
H}}\left({\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})\right)>\frac{d(d+1)}{2}-\sum_{j=1}^{N}\frac{1}{H_{j}},$

due to Corollary 2.1, Lemma 2.2 and the translation invariance of Hausdorff
measure.

Finally, we prove Part (iii) by applying the second part of Lemma 2.3. Notice
that

$\displaystyle{\cal
C}_{l_{1},\ldots,l_{r}}^{\beta}=\left(Y^{\beta}\right)^{-1}\left({\mathbf{S}}(d;l_{1},\ldots,l_{r})\right)\subseteq\left(X^{\beta}\right)^{-1}\left(T_{\beta}^{-1}\left(\mathrm{Im}(G)-A^{\beta}\right)\left(T^{*}_{\beta}\right)^{-1}\right),$

where $(Y^{\beta})^{-1}(B)=\\{t:\,Y^{\beta}(t)\in B\\}$ is the pre-image of
$B$ under the mapping $Y^{\beta}$. We obtain from Lemma 2.3, (3.21), Corollary
2.1 and Lemma 2.2 that almost surely,

$\displaystyle\dim_{{}_{\rm H}}{\cal C}_{l_{1},\ldots,l_{r}}^{\beta}\leq$
$\displaystyle\dim_{{}_{\rm
H}}\left(\left(X^{\beta}\right)^{-1}\left(T_{\beta}^{-1}\left(\mathrm{Im}(G)-A^{\beta}\right)\left(T^{*}_{\beta}\right)^{-1}\right)\cap
I\right)$ $\displaystyle\leq$ $\displaystyle\min_{1\leq\ell\leq
N}\bigg{\\{}\sum_{j=1}^{\ell}\frac{H_{\ell}}{H_{j}}+N-\ell-\dfrac{H_{\ell}}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)\bigg{\\}}.$
(3.24)

In addition, for any $x_{0}\in{\mathbf{S}}(d;l_{1},\ldots,l_{r})$ with with
$|\mathbf{Spec}(x_{0})|=l$, let $\mathfrak{B}_{\delta_{0}}(x_{0})$ be the open
ball in Lemma 3.3. Since

$\displaystyle{\cal C}_{l_{1},\ldots,l_{r}}^{\beta}$
$\displaystyle\supseteq\left(Y^{\beta}\right)^{-1}\left({\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})\right)$
$\displaystyle=\left(X^{\beta}\right)^{-1}\left(T_{\beta}^{-1}\left({\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\right),$

and the Lebesgue measure on
${\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})-A^{\beta}$
satisfies condition (2.6). It follows from Lemma 2.3 that with positive
probability,

$\displaystyle\dim_{{}_{\rm H}}{\cal C}_{l_{1},\ldots,l_{r}}^{\beta}\geq$
$\displaystyle\dim_{{}_{\rm
H}}\left(\left(X^{\beta}\right)^{-1}\left(T_{\beta}^{-1}\left({\mathbf{S}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\right)\right)$
$\displaystyle\geq$ $\displaystyle\min_{1\leq\ell\leq
N}\bigg{\\{}\sum_{j=1}^{\ell}\frac{H_{\ell}}{H_{j}}+N-\ell-\dfrac{H_{\ell}}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)\bigg{\\}}.$
(3.25)

Thus, the first equality in (1.16) follows from (3.1) and (3.1). The second
equality in (1.16) is elementary and can be verified directly. This finishes
the proof of Theorem 1.1. ∎

###  3.2 The complex case: proof of Theorem 1.2

In this subsection, we consider the matrix (1.11) for the case $\beta=2$. The
idea of the proof is similar to that of Section 3.1, and is sketched below.
The following lemma is the complex version of Lemma 3.1, which indicates that
$d^{2}-\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1)$ is an upper bound for the dimension
of ${\mathbf{H}}(d;l_{1},\ldots,l_{r})$.

######  Lemma 3.4.

Let $\Delta:\mathbb{R}^{l}\rightarrow{\mathbf{D}}(d)$ be the function that
maps the vector $u=(u_{1},\ldots,u_{l})\in\mathbb{R}^{l}$ to the diagonal
matrix $\Delta(u)=\\{\Delta_{i,j}(u):1\leq i,j\leq d\\}$ given by (3.1). Then
there exists a compactly supported smooth function
$\widehat{\Gamma}:\mathbb{R}^{d(d-1)-\sum_{j=1}^{r}l_{j}(l_{j}-1)}\rightarrow\mathbf{U}(d)$
such that the mapping
$\widehat{G}:\mathbb{R}^{l}\times\mathbb{R}^{d(d-1)-\sum_{j=1}^{r}l_{j}(l_{j}-1)}\rightarrow{\mathbf{H}}(d)$
given by

$\displaystyle\widehat{G}(u,v)=\widehat{\Gamma}(v)\Delta(u)\widehat{\Gamma}(v)^{*},\quad(u,v)\in\mathbb{R}^{l}\times\mathbb{R}^{d(d-1)-\sum_{j=1}^{r}l_{j}(l_{j}-1)}$
(3.26)

satisfies

$\displaystyle{\mathbf{H}}(d;l_{1},\ldots,l_{r})\subseteq\left\\{x\in\mathbb{R}^{d^{2}}:\
x\in\mathrm{Im}(\widehat{G})\right\\}.$ (3.27)

###### Proof.

The proof is similar to the proof for the real case, and is sketched below. We
consider the set of matrices

$\displaystyle{\mathbf{U}}(d;l_{1},\ldots,l_{r})$
$\displaystyle=\left\\{B=\mathrm{diag}(b_{1},\ldots,b_{l-r},P_{1},\ldots,P_{r})\in{\mathbf{U}}(d):b_{i}\in{\mathbf{U}}(1),P_{j}\in{\mathbf{U}}(l_{j}),\forall
1\leq j\leq r,1\leq i\leq l-r\right\\}.$

equipped with the matrix multiplication forms a Lie subgroup of
${\mathbf{U}}(d)$ with dimension
$l-r+\sum_{j=1}^{r}l_{j}^{2}=d+\sum_{j=1}^{r}l_{j}(l_{j}-1)$. Thus, as the
real case, we define the smooth and free right action of the Lie group
${\mathbf{U}}(d;l_{1},\ldots,l_{r})$ on the manifold ${\mathbf{U}}(d)$ as
matrix multiplication. Moreover, the right action is proper by Lemma 2.8.
Therefore, by Lemma 2.9 and Lemma 2.6, the quotient space
${\mathbf{U}}(d)/{\mathbf{U}}(d;l_{1},\ldots,l_{r})$ is a smooth manifold of
dimension $d(d-1)-\sum_{j=1}^{r}l_{j}(l_{j}-1)$, and the quotient map
$\pi:{\mathbf{U}}(d)\longrightarrow{\mathbf{U}}(d)/{\mathbf{U}}(d;l_{1},\ldots,l_{r})$
is a smooth submersion and is an open map.

For $\epsilon>0$, let $I_{\epsilon}$ be the open interval

$\displaystyle
I_{\epsilon}=(-\epsilon,\epsilon)^{d(d-1)-\sum_{j=1}^{r}l_{j}(l_{j}-1)}.$

Then for any $A\in{\mathbf{U}}(d)$, one can define the a smooth diffeomorphism
$\phi$ from $I_{\epsilon}$ to a neighbourhood of $\pi(A)$ such that
$\phi(0)=\pi(A)$. Then we can define $\Gamma^{(i)}$ on $I_{\epsilon}$ as in
(3.7), and $G^{(i)}$ as in (3.9). By the compactness of
${\mathbf{U}}(d)/{\mathbf{U}}(d;l_{1},\ldots,l_{r})$, we can show that
${\mathbf{U}}(d;l_{1},\ldots,l_{r})$ is covered by finite many
$\mathrm{Im}(G^{(i)})$. Thus, we can construct $\Gamma$ as the end of the
proof of Lemma 3.1. ∎

The following result is the complex version of Lemma 3.2. The proof is similar
to that of Lemma 3.2 and hence is omitted.

######  Lemma 3.5.

Let $A\in{\mathbf{H}}(d;l_{1},\ldots,l_{r})$ be a Hermitian matrix with
spectral decomposition $A=PDP^{*}$ for some $P\in{\mathbf{U}}(d)$ and
$D\in{\mathbf{D}}(d)$ such that $|\mathbf{Spec}(A)|=l$. Then for every
$\epsilon>0$, there exists $\delta>0$, such that for all
$B\in{\mathbf{H}}(d;l_{1},\ldots,l_{r})$ satisfying

$\displaystyle\max_{1\leq i,j\leq d}|A_{i,j}-B_{i,j}|<\delta,$

we have $|\mathbf{Spec}(B)|=l$, and there exists a spectral decomposition of
the form $B=QFQ^{*}$, where $Q\in{\mathbf{U}}(d)$ and $F\in{\mathbf{D}}(d)$
satisfy

$\displaystyle\max_{1\leq i\leq d}|D_{i,i}-F_{i,i}|<\epsilon,\quad\max_{1\leq
i,j\leq d}|Q_{i,j}-P_{i,j}|<\epsilon.$

The next lemma is the complex version of Lemma 3.3, which indicates that
$d^{2}-\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1)$ is an optimal upper bound for the
dimension of ${\mathbf{H}}(d;l_{1},\ldots,l_{r})$. The proof is similar to the
proof of Lemma 3.3, and hence is omitted.

######  Lemma 3.6.

For $x_{0}\in{\mathbf{H}}(d;l_{1},\ldots,l_{r})$ with
$|\mathbf{Spec}(x_{0})|=l$, there exists $\delta_{0}>0$, such that
${\mathbf{H}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$ is a
$(d^{2}-\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1))$-dimensional manifold with positive
$(d^{2}-\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1))$-dimensional Lebesgue measure.

Now we are ready to prove Theorem 1.2.

###### Proof of Theorem 1.2.

The proof is similar to that of Theorem 1.1, which is sketched below.

We first prove Part (i). By Lemma 3.4, there exists a smooth map

$\displaystyle\widehat{G}:\mathbb{R}^{l}\times\mathbb{R}^{d(d-1)-\sum_{j=1}^{r}l_{j}(l_{j}-1)}\rightarrow{\mathbf{H}}(d),$

such that
${\mathbf{H}}(d;l_{1},\ldots,l_{r})\subseteq\mathrm{Im}(\widehat{G})$.
Together with Lemma 3.6, we have

$\displaystyle\dim_{{}_{\rm
H}}\left(\mathrm{Im}\left(\widehat{G}\right)\right)=d^{2}-\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1).$
(3.28)

If

$\displaystyle\sum_{j=1}^{N}\dfrac{1}{H_{j}}<\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1),$

then by Lemma 2.3, the translation invariance of Hausdorff measure, (3.28),
Corollary 2.1 and Lemma 2.2, we derive that

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)$
$\displaystyle=\mathbb{P}\left(Y^{\beta}(t)\in{\mathbf{H}}(d;l_{1},\ldots,l_{r})\mathrm{\
for\ some\ }t\in I\right)$ $\displaystyle\leq\mathbb{P}\left(X^{\beta}(I)\cap
T_{\beta}^{-1}\left(\mathrm{Im}\left(\widehat{G}\right)-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\not=\emptyset\right)$
$\displaystyle\leq
c_{6}{\mathbf{H}}_{d^{2}-\sum_{j=1}^{N}\frac{1}{H_{j}}}\left(T_{\beta}^{-1}\left(\mathrm{Im}\left(\widehat{G}\right)-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\right)=0.$

If

$\displaystyle\sum_{j=1}^{N}\dfrac{1}{H_{j}}=\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1),$

then we can compute the Lebesgue measure via Lemma 2.5

$\displaystyle\lambda_{d^{2}}\left(\left(\mathrm{Im}\left(\widehat{G}\big{|}_{F_{K}}\right)-A^{\beta}\right)^{(r)}\right)\leq
Cr^{\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1)}=Cr^{\sum_{j=1}^{N}\frac{1}{H_{j}}}$

for all $r>0$ small. Here, $F_{K}$ is a compact interval of
$\mathbb{R}^{d^{2}-\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1)}$ given by

$\displaystyle F_{K}=[-K,K]^{l}\times F^{\prime},$

where $F^{\prime}$ is a compact interval containing the support of
$\widehat{\Gamma}$ given by Lemma 3.4. Then by Lemma 2.4 and Corollary 2.2,
for $K$ large, we have

$\displaystyle\mathbb{P}\bigg{(}\left\\{\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right\\}$
$\displaystyle\quad\quad\quad\quad\quad\quad\cap\left\\{\mathbf{Spec}\left(Y^{\beta}\right)\subseteq[-K,K]\right\\}\bigg{)}$
$\displaystyle\leq\mathbb{P}\left(X^{\beta}(I)\cap
T_{\beta}^{-1}\left(\mathrm{Im}\left(\widehat{G}\big{|}_{F_{K}}\right)-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\not=\emptyset\right)=0.$

Letting $K\to\infty$, we obtain

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)=0.$

Next, we prove (ii). We choose $x_{0}\in{\mathbf{H}}(d;l_{1},\ldots,l_{r})$
satisfying $\mathbf{Spec}(x_{0})=l$. By Lemma 3.6, there exists
$\delta_{0}>0$, such that
${\mathbf{H}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$ is an
$(d^{2}-\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1))$-dimensional manifold. Thus, by
Lemma 2.3, when

$\displaystyle\sum_{j=1}^{N}\dfrac{1}{H_{j}}>\sum_{j=1}^{r}(l_{j}+1)(l_{j}-1),$

we have

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|\lambda_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)$
$\displaystyle=\mathbb{P}\left(X^{\beta}(t)\in
T_{\beta}^{-1}\left({\mathbf{H}}(d;l_{1},\ldots,l_{r})-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\mathrm{\
for\ some\ }t\in I\right)$ $\displaystyle\geq\mathbb{P}\left(X^{\beta}(I)\in
T_{\beta}^{-1}\left({\mathbf{H}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\not=\emptyset\right)$
$\displaystyle\geq
c_{5}\mathcal{C}_{d^{2}-\sum_{j=1}^{N}\frac{1}{H_{j}}}\left(T_{\beta}^{-1}\left({\mathbf{H}}(d;l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})-A^{\beta}\right)\left(T_{\beta}^{*}\right)^{-1}\right)>0,$

where we use Corollary 2.1 and Lemma 2.2.

The proof of (iii), which is an application of Lemma 2.3, is similar to the
proof of part (iii) of Theorem 1.1 and is omitted. This finishes the proof. ∎

##  4 Multiple collisions of singular value processes

###  4.1 The real case: proof of Theorem 1.3

Fix $r\in\mathbb{N}$, $l_{1},\ldots,l_{r}\in\\{1,\ldots,d_{1}\\}$ satisfying
$\sum_{j=1}^{r}l_{j}\leq d_{1}$. For simplicity, we denote
$l=d_{1}-\sum_{j=1}^{r}(l_{j}-1)$ be the number of distinct singular values,
and $L_{i}=d_{1}-\sum_{j=i}^{r}l_{j}+1$. Then one can check that
$L_{1}=l-r+1$. We also denote $l_{0}=d_{2}-d_{1}$ be the number of trivial
singular values.

We start with an upper bound for the dimension of
${\mathbf{M}}_{\mathbb{R}}(d_{1}\times d_{2};l_{1},\ldots,l_{r})$ given in
(1.4). Lemma 4.1 below implies that
$d_{1}d_{2}-\frac{1}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)$ is an upper bound of
the dimension of ${\mathbf{M}}_{\mathbb{R}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})$.

######  Lemma 4.1.

Let $\Delta:\mathbb{R}_{\geq 0}^{l}\rightarrow{\mathbf{D}}(d_{1}\times d_{2})$
be a function that maps each vector $u=(u_{1},\ldots,u_{l})\in\mathbb{R}_{\geq
0}^{l}$ to a $d_{1}\times d_{2}$ diagonal matrix $\Delta(u)$ with entries
given by

$\displaystyle\Delta_{i,i}(u)=\begin{cases}u_{i},&1\leq i\leq L_{1}-1;\\\
u_{l-r+j},&L_{j}\leq i\leq L_{j}+l_{j}-1.\end{cases}$

Then there exist compactly supported smooth functions
$\Gamma_{L}:\mathbb{R}^{\frac{1}{2}[d_{2}(d_{2}-1)+d_{1}(d_{1}-1)-\sum_{j=0}^{r}l_{j}(l_{j}-1)]}\rightarrow\mathbf{O}(d_{1})$
and
$\Gamma_{R}:\mathbb{R}^{\frac{1}{2}[d_{2}(d_{2}-1)+d_{1}(d_{1}-1)-\sum_{j=0}^{r}l_{j}(l_{j}-1)]}\rightarrow\mathbf{O}(d_{2})$,
such that the mapping

$\displaystyle G:\mathbb{R}_{\geq
0}^{l}\times\mathbb{R}^{\frac{1}{2}[d_{2}(d_{2}-1)+d_{1}(d_{1}-1)-\sum_{j=0}^{r}l_{j}(l_{j}-1)]}\rightarrow\mathbb{R}^{d_{1}\times
d_{2}}$

given by

$\displaystyle
G(u,v)=\Gamma_{L}(v)\Delta(u)\Gamma_{R}(v)^{*},~{}(u,v)\in\mathbb{R}_{\geq
0}^{l}\times\mathbb{R}^{\frac{1}{2}[d_{2}(d_{2}-1)+d_{1}(d_{1}-1)-\sum_{j=0}^{r}l_{j}(l_{j}-1)]},$
(4.1)

satisfies

$\displaystyle{\mathbf{M}}_{\mathbb{R}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})\subseteq\left\\{x\in\mathbb{R}^{d_{1}\times d_{2}}:\
x\in\mathrm{Im}(G)\right\\},$

where $\mathrm{Im}(G)$ is the image of the mapping $G$.

###### Proof.

Recall that $\mathbf{O}(d_{1})$ and $\mathbf{O}(d_{2})$ are Lie groups of
dimension $d_{1}(d_{1}-1)/2$ and $d_{2}(d_{2}-1)/2$ respectively. Thus, by
[Lee2013]*Example 7.3 (k), their direct product
$\mathbf{O}(d_{1})\times\mathbf{O}(d_{2})$ is again a Lie group with dimension
$d_{1}(d_{1}-1)/2+d_{2}(d_{2}-1)/2$. Recall the definition (3.1) of the Lie
group $\mathbf{O}(d_{2};l_{1},\ldots,l_{r},l_{0})$, then for any
$g\in\mathbf{O}(d_{2};l_{1},\ldots,l_{r},l_{0})$, one can write
$g=\mathrm{diag}(g_{d_{1}},g_{l_{0}})$ with
$g_{d_{1}}\in\mathbf{O}(d_{1};l_{1},\ldots,l_{r})$ and
$g_{l_{0}}\in\mathbf{O}(l_{0})$. If $l_{0}=0$, then we use the convention that
$\mathbf{O}(d_{2};l_{1},\ldots,l_{r},l_{0})=\mathbf{O}(d_{1};l_{1},\ldots,l_{r})$
and in this case we write $g=g_{d_{1}}$. Then one can verify that the map

$\displaystyle\left(\mathbf{O}(d_{1})\times\mathbf{O}(d_{2})\right)\times\mathbf{O}(d_{2};l_{1},\ldots,l_{r},l_{0})\to\mathbf{O}(d_{1})\times\mathbf{O}(d_{2})$

given by

$\displaystyle((p_{1},p_{2}),g)\mapsto(p_{1},p_{2})\cdot
g=(p_{1}g_{d_{1}},p_{2}g).$

is the right action of the Lie group
$\mathbf{O}(d_{2};l_{1},\ldots,l_{r},l_{0})$ on the manifold
$\mathbf{O}(d_{1})\times\mathbf{O}(d_{2})$. One can deduce the smoothness of
the right action from the smoothness of the matrix multiplication. Besides,
the right action is free since identity matrix is the only multiplication
identity element. In addition, by Lemma 2.8 and the compactness of
$\mathbf{O}(d_{2};l_{1},\ldots,l_{r},l_{0})$, the right action is proper.
Therefore, by Lemma 2.9, the orbit space
$\left(\mathbf{O}(d_{1})\times\mathbf{O}(d_{2})\right)/\mathbf{O}(d_{2};l_{1},\ldots,l_{r},l_{0})$
is a smooth manifold of dimension
$\frac{1}{2}[d_{2}(d_{2}-1)+d_{1}(d_{1}-1)-\sum_{j=0}^{r}l_{j}(l_{j}-1)]$, and
the quotient map
$\pi:\mathbf{O}(d_{1})\times\mathbf{O}(d_{2})\longrightarrow\left(\mathbf{O}(d_{1})\times\mathbf{O}(d_{2})\right)/\mathbf{O}(d_{2};l_{1},\ldots,l_{r},l_{0})$
is a smooth submersion. Furthermore, $\pi$ is an open map by Lemma 2.6.

For any $p\in\mathbf{O}(d_{1})\times\mathbf{O}(d_{2})$, since $\pi$ is a
submersion, by Lemma 2.7, there exists a smooth local section $\sigma_{\pi}$
from an open neighbourhood of $\pi(p)$ to
$\mathbf{O}(d_{1})\times\mathbf{O}(d_{2})$ that maps $\pi(p)$ to $p$ and that
$\pi\circ\sigma_{\pi}$ is identity. Moreover, by the definition of chart (see
e.g. [Tu2011]*Definition 5.1), there exists an open neighbourhood of $\pi(p)$
in
$\left(\mathbf{O}(d_{1})\times\mathbf{O}(d_{2})\right)/\mathbf{O}(d_{2};l_{1},\ldots,l_{r},l_{0})$
that is smoothly diffeomorphic to $I_{\epsilon}$, which is defined by

$\displaystyle
I_{\epsilon}=(-\epsilon,\epsilon)^{\frac{1}{2}[d_{2}(d_{2}-1)+d_{1}(d_{1}-1)-\sum_{j=0}^{r}l_{j}(l_{j}-1)]}.$

That is, there exist $\delta>0$ and a diffeomorphism

$\displaystyle\phi:I_{\epsilon}\to\phi(I_{\epsilon})\subset\left(\mathbf{O}(d_{1})\times\mathbf{O}(d_{2})\right)/\mathbf{O}(d_{2};l_{1},\ldots,l_{r},l_{0})\cap\pi(\mathfrak{B}_{\delta}(p))$

satisfying $\phi(0)=\pi(p)$. Then we construct smooth function

$\displaystyle\Gamma=(\Gamma_{L},\Gamma_{R}):\mathbb{R}^{\frac{1}{2}[d_{2}(d_{2}-1)+d_{1}(d_{1}-1)-\sum_{j=0}^{r}l_{j}(l_{j}-1)]}\rightarrow{\mathbf{O}}(d_{1})\times{\mathbf{O}}(d_{2})$

that is supported on $I_{2\epsilon}$ and satisfies

$\displaystyle(\Gamma_{L}(v),\Gamma_{R}(v))=\Gamma(v)=\sigma_{\pi}\circ\phi(v),\forall
v\in I_{\epsilon}.$

Since the Lie groups $\mathbf{O}(d_{1})$ and $\mathbf{O}(d_{2})$ are compact,
by Tychonoff’s theorem, the product space
$\mathbf{O}(d_{1})\times\mathbf{O}(d_{2})$ is also compact, which yields the
compactness of the quotient space
$\left(\mathbf{O}(d_{1})\times\mathbf{O}(d_{2})\right)/\mathbf{O}(d_{2};l_{1},\ldots,l_{r},l_{0})$.
As $\phi$ is a diffeomorphism, $\phi(I_{\epsilon})$ is an open subset of the
quotient space
$\left(\mathbf{O}(d_{1})\times\mathbf{O}(d_{2})\right)/\mathbf{O}(d_{2};l_{1},\ldots,l_{r},l_{0})$
containing $\pi(p)$. Hence, when $p$ runs over
$\mathbf{O}(d_{1})\times\mathbf{O}(d_{2})$, the collection of the sets
$\phi(I_{\epsilon})$ forms an open cover of
$\left(\mathbf{O}(d_{1})\times\mathbf{O}(d_{2})\right)/\mathbf{O}(d_{2};l_{1},\ldots,l_{r},l_{0})$.
By the compactness, there exists a finite subcover
$\left\\{\phi^{(i)}(I_{\epsilon_{i}}):i=1,\ldots,M\right\\}$ for some
$M\in\mathbb{N}$.

For $1\leq i\leq M$, we define mappings $G^{(i)}:\mathbb{R}_{\geq 0}^{l}\times
I_{\epsilon_{i}}\rightarrow\mathbb{R}^{d_{1}\times d_{2}}$ by

$\displaystyle
G^{(i)}(u,v)=\Gamma_{L}^{(i)}(v)\Delta(u)\Gamma_{R}^{(i)}(v)^{*},$

for $u\in\mathbb{R}_{\geq 0}^{l}$, $v\in I_{\epsilon_{i}}$.

For an arbitrary fixed $x\in{\mathbf{M}}_{\mathbb{R}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})$, we have the singular value decomposition
$x=P_{L}DP_{R}^{*}$ for some $P_{L}\in{\mathbf{O}}(d_{1})$,
$P_{R}\in{\mathbf{O}}(d_{2})$ and $D\in{\mathbf{D}}(d_{1}\times d_{2})$. We
assume that $D_{L_{j},L_{j}}=\cdots=D_{L_{j}+l_{j}-1,L_{j}+l_{j}-1}$ for
$1\leq j\leq r$, by rearranging the diagonal of $D$, the columns of $P_{L}$
and $P_{R}$ if necessary. There exists $i_{0}\in\\{1,\dots,M\\}$ such that
$\pi(P_{L},P_{R})\in\phi^{(i_{0})}\left(I_{\epsilon_{i_{0}}}\right)$, and
hence one can find $v\in I_{\epsilon_{i_{0}}}$ such that
$\pi(P_{L},P_{R})=\phi^{(i_{0})}(v)$. Thus,
$\left(\Gamma_{L}^{(i_{0})}(v),\Gamma_{R}^{(i_{0})}(v)\right)=\sigma_{\pi}\circ\pi(P_{L},P_{R})$.
Note that $\sigma_{\pi}\circ\pi(P_{L},P_{R})$ and $(P_{L},P_{R})$ are in the
same orbit space under the right action of the Lie group
${\mathbf{O}}(d_{1};l_{1},\ldots,l_{r},l_{0})$, we have
$\sigma_{\pi}\circ\pi(P_{L},P_{R})=(P_{L},P_{R})\cdot
g=(P_{L}g_{d_{1}},P_{R}g)$, where $g=\mathrm{diag}(g_{d_{1}},g_{l_{0}})$ is an
element in the Lie group ${\mathbf{O}}(d_{1};l_{1},\ldots,l_{r},l_{0})$. Let
$u=(D_{1,1},D_{2,2},\ldots,D_{L_{1}-1,L_{1}-1},D_{L_{1},L_{1}},D_{L_{2},L_{2}},\ldots,D_{L_{r},L_{r}})\in\mathbb{R}_{\geq
0}^{l}$ then $D=\Delta(u)$. Thus, $x$ has the decomposition

$\displaystyle
x=P_{L}DP_{R}^{*}=P_{L}\left(g_{d_{1}}Dg^{*}\right)P_{R}^{*}=(P_{L}\cdot
g_{d_{1}})D(P_{R}\cdot
g)^{*}=\Gamma_{L}^{(i_{0})}(v)\Delta(u)\Gamma_{R}^{(i_{0})}(v)^{*}=G^{(i_{0})}(u,v).$

Since $x\in{\mathbf{M}}_{\mathbb{R}}(d_{1}\times d_{2};l_{1},\ldots,l_{r})$ is
arbitrarily chosen, we conclude that

$\displaystyle{\mathbf{M}}_{\mathbb{R}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})\subseteq\left\\{x\in\mathbb{R}^{d_{1}\times
d_{2}}:x\in\bigcup_{i=1}^{M}\mathrm{Im}(G^{(i)})\right\\}.$

Finally, let $\epsilon=\max\left\\{\epsilon_{1},\ldots,\epsilon_{M}\right\\}$,
then we can construct smooth functions $\Gamma_{L}$ and $\Gamma_{R}$ with
support $I_{4M\epsilon}$ as what we do at the end of the proof of Lemma 3.1. ∎

The following lemma is about the continuity of the singular value
decomposition on ${\mathbf{M}}_{\mathbb{R}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})$.

######  Lemma 4.2.

Let $A\in{\mathbf{M}}_{\mathbb{R}}(d_{1}\times d_{2};l_{1},\ldots,l_{r})$ with
singular value decomposition $A=P_{L}DP_{R}^{*}$ for some
$P_{L}\in{\mathbf{O}}(d_{1})$, $P_{R}\in{\mathbf{O}}(d_{2})$ and
$D\in{\mathbf{D}}(d_{1}\times d_{2})$, such that
$0\notin\mathbf{Spec}(AA^{*})$ and $|\mathbf{Spec}(AA^{*})|=l$. Then for any
$\epsilon>0$, there exists $\delta>0$, such that for all
$B\in{\mathbf{M}}_{\mathbb{R}}(d_{1}\times d_{2};l_{1},\ldots,l_{r})$
satisfying

$\displaystyle\max_{1\leq i\leq d_{1},1\leq j\leq
d_{2}}|A_{i,j}-B_{i,j}|<\delta,$

we have $0\notin\mathbf{Spec}(BB^{*})$ and $|\mathbf{Spec}(BB^{*})|=l$.
Moreover, there exists a singular value decomposition $B=Q_{L}FQ_{R}^{*}$,
where $Q_{L}\in{\mathbf{O}}(d_{1})$, $Q_{R}\in{\mathbf{O}}(d_{2})$ and
$F\in{\mathbf{D}}(d_{1}\times d_{2})$ satisfy

$\displaystyle\max_{1\leq i\leq
d_{1}}|D_{i,i}-F_{i,i}|<\epsilon,\quad\max_{1\leq i,j\leq
d_{1}}\left|\left(Q_{L}\right)_{i,j}-\left(P_{L}\right)_{i,j}\right|<\epsilon,\quad\max_{1\leq
i,j\leq
d_{2}}\left|\left(Q_{R}\right)_{i,j}-\left(P_{R}\right)_{i,j}\right|<\epsilon.$
(4.2)

###### Proof.

Noting that $A\in{\mathbf{M}}_{\mathbb{R}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})$ and $|\mathbf{Spec}(AA^{*})|=l$, without loss of
generality we assume

$\displaystyle
0<D_{1,1}<\cdots<D_{L_{1},L_{1}}=\cdots=D_{L_{2}-1,L_{2}-1}<D_{L_{2},L_{2}}=\cdots=D_{L_{3}-1,L_{3}-1}<\cdots<D_{L_{r},L_{r}}=\cdots=D_{d,d}.$
(4.3)

As $0\notin\mathbf{Spec}(AA^{*})$, we have
$A^{*}A\in{\mathbf{S}}(d_{2};l_{1},\ldots,l_{r},l_{0})$ with decomposition
$A^{*}A=P_{R}D^{*}DP_{R}^{*}$. Moreover, for
$B\in{\mathbf{M}}_{\mathbb{R}}(d_{1}\times d_{2};l_{1},\ldots,l_{r})$
satisfying

$\displaystyle\max_{1\leq i\leq d_{1},1\leq j\leq
d_{2}}|A_{i,j}-B_{i,j}|<\delta,$

we have $B^{*}B\in{\mathbf{S}}(d_{2};l_{1},\ldots,l_{r},l_{0})$ and

$\displaystyle\max_{1\leq i,j\leq
d_{2}}\left|(A^{*}A)_{i,j}-(B^{*}B)_{i,j}\right|$ $\displaystyle=\max_{1\leq
i,j\leq
d_{2}}\left|\sum_{k=1}^{d_{1}}A_{i,k}^{*}A_{k,j}-\sum_{k=1}^{d_{1}}B_{i,k}^{*}B_{k,j}\right|$
$\displaystyle=\max_{1\leq i,j\leq
d_{2}}\left|\sum_{k=1}^{d_{1}}(A_{i,k}^{*}-B_{i,k}^{*})A_{k,j}+\sum_{k=1}^{d_{1}}B_{i,k}^{*}(A_{k,j}-B_{k,j})\right|$
$\displaystyle<\delta\left(d_{1}\max_{1\leq i\leq d_{1},1\leq j\leq
d_{2}}\left|A_{i,j}\right|+d_{1}\max_{1\leq i\leq d_{1},1\leq j\leq
d_{2}}\left|B_{i,j}\right|\right)$
$\displaystyle<\delta\left(2d_{1}\max_{1\leq i\leq d_{1},1\leq j\leq
d_{2}}\left|A_{i,j}\right|+d_{1}\delta\right).$

Thus, by Lemma 3.2, for any $\epsilon>0$, we can choose $\delta$ small enough,
so that $|\mathbf{Spec}(B^{*}B)|=l+1$ if $l_{0}>0$ and
$|\mathbf{Spec}(B^{*}B)|=l$ if $l_{0}=0$, and there exists a spectral
decomposition $B^{*}B=Q_{R}F_{R}Q_{R}^{*}$, where $Q_{R}\in\mathbf{O}(d_{2})$
and $F_{R}\in\mathbf{D}(d_{2})$ satisfy

$\displaystyle\max_{1\leq i\leq
d_{2}}\left|\left(D^{*}D\right)_{i,i}-\left(F_{R}\right)_{i,i}\right|<\epsilon,\quad\max_{1\leq
i,j\leq
d_{2}}\left|\left(Q_{R}\right)_{i,j}-\left(P_{R}\right)_{i,j}\right|<\epsilon.$
(4.4)

We choose $\epsilon$ to be small enough, then by (4.3), we have

$\displaystyle
0<(F_{R})_{1,1}<\cdots<(F_{R})_{L_{1},L_{1}}<(F_{R})_{L_{2},L_{2}}<\cdots<(F_{R})_{L_{r},L_{r}},\
\left|(F_{R})_{d_{1}+1,d_{1}+1}\right|,\ldots,\left|(F_{R})_{d_{2},d_{2}}\right|<\epsilon.$

Since $B^{*}B\in{\mathbf{S}}(d_{2};l_{1},\ldots,l_{r},l_{0})$, we have
$(F_{R})_{i,i}=(F_{R})_{L_{j},L_{j}}$ for $L_{j}\leq i\leq L_{j}+l_{j}-1$,
$1\leq j\leq r$, and $(F_{R})_{i,i}=0$ for $d_{1}+1\leq i\leq d_{2}$.

Now consider the singular value decomposition
$B=Q_{L}^{\prime}FQ_{R}^{\prime*}$ for some
$Q_{L}^{\prime}\in{\mathbf{O}}(d_{1})$, $Q_{R}^{\prime}\in{\mathbf{O}}(d_{2})$
and $F\in{\mathbf{D}}(d_{1}\times d_{2})$. By rearranging the diagonal of $F$,
the columns of $Q_{L}^{\prime}$ and $Q_{R}^{\prime}$ if necessary, we can
assume that the diagonal entries of $F$ are in a non-decreasing order. Then we
have the spectral decomposition $B^{*}B=Q_{R}^{\prime}F^{*}FQ_{R}^{\prime*}$.
Note that both $F^{*}F$ and $F_{R}$ are diagonal matrices of eigenvalues of
$B^{*}B$ whose last $d_{2}-d_{1}$ diagonal entries are zero and whose first
$d_{1}$ entries are in a non-decreasing order, we have $F^{*}F=F_{R}$. Hence,
we have

$\displaystyle
0<F_{1,1}<\cdots<F_{L_{1},L_{1}}=\cdots=F_{L_{2}-1,L_{2}-1}<F_{L_{2},L_{2}}=\cdots=F_{L_{3}-1,L_{3}-1}<\cdots<F_{L_{r},L_{r}}=\cdots=F_{d_{1},d_{1}}.$
(4.5)

For $1\leq i\leq d_{2}$, the $i$th columns of $Q_{R}$ and $Q_{R}^{\prime}$ are
the eigenvectors of $B^{*}B$ associated with the eigenvalue
$\left(F_{R}\right)_{i,i}$. Therefore, there exists
$g\in\mathbf{O}(d_{2};l_{1},\ldots,l_{r},l_{0})$, such that
$Q_{R}^{\prime}=Q_{R}\cdot g$. Moreover, the $d_{2}\times d_{2}$ matrix $g$
has the form $g=\mathrm{diag}\left(g_{d_{1}},g_{l_{0}}\right)$, where
$g_{d_{1}}\in\mathbf{O}(d_{1};l_{1},\ldots,l_{r})$ and
$g_{l_{0}}\in\mathbf{O}(l_{0})$. One can check that $g_{d_{1}}Fg^{*}=F$. Let
$Q_{L}=Q_{L}^{\prime}g_{d_{1}}^{*}$ then $Q_{L}\in\mathbf{O}(d_{1})$.
Therefore, we have

$\displaystyle
B=Q_{L}^{\prime}FQ_{R}^{\prime*}=Q_{L}^{\prime}\left(g_{d_{1}}^{*}g_{d_{1}}\right)F\left(g^{*}g\right)Q_{R}^{\prime*}=\left(Q_{L}^{\prime}g_{d_{1}}^{*}\right)\left(g_{d_{1}}Fg^{*}\right)\left(Q_{R}^{\prime}g^{*}\right)^{*}=Q_{L}FQ_{R}^{*}.$
(4.6)

It remains to verify (4.2) for the singular value decomposition (4.6). The
last inequality in (4.2) follows directly from (4.4).

Let $K$ be a large number such that

$\displaystyle\max_{i\in[d_{1}],j\in[d_{2}]}|A_{i,j}|<K,\ D_{1,1}>2/K.$

By triangle inequality, one can easily deduce that
$\left|B_{i,j}\right|<K+\delta$ for all $i\in[d_{1}],j\in[d_{2}]$. Without
loss of generality, we may assume that $\epsilon\ll 1/K^{2}$, then by (4.4),
we have

$\displaystyle F_{i,i}^{2}\geq
D_{i,i}^{2}-\left|F_{i,i}^{2}-D_{i,i}^{2}\right|\geq
D_{1,1}^{2}-\epsilon>1/K^{2}.$

Hence, by (4.4), we have

$\displaystyle\max_{1\leq i\leq d_{1}}|D_{i,i}-F_{i,i}|=\max_{1\leq i\leq
d_{1}}\dfrac{\left|D_{i,i}^{2}-F_{i,i}^{2}\right|}{D_{i,i}+F_{i,i}}<\dfrac{K}{3}\max_{1\leq
i\leq d_{1}}\left|D_{i,i}^{2}-F_{i,i}^{2}\right|<\dfrac{K\epsilon}{3}.$ (4.7)

Note that $P_{R}$ and $Q_{R}$ are orthogonal matrices, by triangle inequality
and (4.4), we have

$\displaystyle\max_{i\in[d_{1}],j\in[d_{2}]}\left|\left(Q_{L}F\right)_{i,j}-\left(P_{L}D\right)_{i,j}\right|=\max_{i\in[d_{1}],j\in[d_{2}]}\left|\left(BQ_{R}\right)_{i,j}-\left(AP_{R}\right)_{i,j}\right|$
$\displaystyle=\max_{i\in[d_{1}],j\in[d_{2}]}\left|\sum_{k=1}^{d_{2}}B_{i,k}\left(Q_{R}\right)_{k,j}-\sum_{k=1}^{d_{2}}A_{i,k}\left(P_{R}\right)_{k,j}\right|$
$\displaystyle=\max_{i\in[d_{1}],j\in[d_{2}]}\left|\sum_{k=1}^{d_{2}}B_{i,k}\left(\left(Q_{R}\right)_{k,j}-\left(P_{R}\right)_{k,j}\right)+\sum_{k=1}^{d_{2}}\left(B_{i,k}-A_{i,k}\right)\left(P_{R}\right)_{k,j}\right|$
$\displaystyle<d_{2}\epsilon(K+\delta)+d_{2}\delta.$ (4.8)

Note that the matrices $D,F\in{\mathbf{D}}(d_{1}\times d_{2})$, we can write
$D=\left(D_{d_{1}},0_{d_{1}\times l_{0}}\right)$ and
$F=\left(F_{d_{1}},0_{d_{1}\times l_{0}}\right)$, where
$D_{d_{1}},F_{d_{1}}\in{\mathbf{D}}(d_{1})$, and $0_{d_{1}\times
l_{0}}\in\mathbb{R}^{d_{1}\times l_{0}}$ is a zero matrix. Then we have
$P_{L}D=\left(P_{L}D_{d_{1}},0_{d_{1}\times l_{0}}\right)$ and
$Q_{L}F=\left(Q_{L}F_{d_{1}},0_{d_{1}\times l_{0}}\right)$. As $D_{d_{1}}$ and
$F_{d_{1}}$ are diagonal square matrices with strictly positive diagonal
entries, they are invertible. Hence, by triangle inequality, (4.7) and (4.1),

$\displaystyle\max_{i,j\in[d_{1}]}\left|\left(Q_{L}\right)_{i,j}-\left(P_{L}\right)_{i,j}\right|=\max_{i,j\in[d_{1}]}\left|\left(Q_{L}F_{d_{1}}F_{d_{1}}^{-1}\right)_{i,j}-\left(P_{L}D_{d_{1}}D_{d_{1}}^{-1}\right)_{i,j}\right|$
$\displaystyle\leq\max_{i,j\in[d_{1}]}\left|\left(Q_{L}F_{d_{1}}\right)_{i,j}\left(F_{d_{1}}^{-1}\right)_{j,j}-\left(P_{L}D_{d_{1}}\right)_{i,j}\left(D_{d_{1}}^{-1}\right)_{j,j}\right|$
$\displaystyle\leq\max_{i,j\in[d_{1}]}\left|\left(\left(Q_{L}F_{d_{1}}\right)_{i,j}-\left(P_{L}D_{d_{1}}\right)_{i,j}\right)\left(F_{d_{1}}^{-1}\right)_{j,j}+\left(P_{L}D_{d_{1}}\right)_{i,j}\left(\left(F_{d_{1}}^{-1}\right)_{j,j}-\left(D_{d_{1}}^{-1}\right)_{j,j}\right)\right|$
$\displaystyle\leq\max_{i,j\in[d_{1}]}\left|\left(\left(Q_{L}F_{d_{1}}\right)_{i,j}-\left(P_{L}D_{d_{1}}\right)_{i,j}\right)\left(F_{d_{1}}^{-1}\right)_{j,j}+\left(P_{L}\right)_{i,j}\left(\left(D_{d_{1}}\right)_{j,j}-\left(F_{d_{1}}\right)_{j,j}\right)\left(F_{d_{1}}^{-1}\right)_{j,j}\right|$
$\displaystyle<\left(d_{2}\epsilon(K+\delta)+d_{2}\delta\right)K+\dfrac{K^{2}\epsilon}{3}.$
(4.9)

Therefore, the proof is concluded from (4.4), (4.7) and (4.1). ∎

The next lemma implies that the upper bounded for
${\mathbf{M}}_{\mathbb{R}}(d_{1}\times d_{2};l_{1},\ldots,l_{r})$ given by
Lemma 4.1 is optimal.

######  Lemma 4.3.

For $x_{0}\in{\mathbf{M}}_{\mathbb{R}}(d_{1}\times d_{2};l_{1},\ldots,l_{r})$
with $0\notin\mathbf{Spec}(x_{0}x_{0}^{*})$ and
$|\mathbf{Spec}(x_{0}x_{0}^{*})|=l$, there exists $\delta_{0}>0$ such that
${\mathbf{M}}_{\mathbb{R}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$ is a
$(d_{1}d_{2}-\frac{1}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2))$-dimensional
manifold. In particular, ${\mathbf{M}}_{\mathbb{R}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$ has positive
$(d_{1}d_{2}-\frac{1}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2))$-dimensional
Lebesgue measure.

###### Proof.

By Lemma 4.1, there exist $u_{0}\in\mathbb{R}_{+}^{l}$,
$v_{0}\in\mathbb{R}^{\frac{1}{2}[d_{2}(d_{2}-1)+d_{1}(d_{1}-1)-\sum_{j=0}^{r}l_{j}(l_{j}-1)]}$,
such that $x_{0}=G(u_{0},v_{0})$, where the function $G$ is given by (4.1).
Moreover, by the construction, without loss of generality, we may assume that
$x_{0}=G^{(1)}(u_{0},v_{0})=\Gamma_{L}^{(1)}(v_{0})\Delta(u_{0})\Gamma_{R}^{(1)}(v_{0})$,
with $v_{0}\in I_{\epsilon_{1}}$.

To show that the manifold ${\mathbf{M}}_{\mathbb{R}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$ has dimension
$d_{1}d_{2}-\frac{1}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)$ and has positive
$(d_{1}d_{2}-\frac{1}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2))$-dimensional
Lebesgue measure, it is sufficient to show that there exist open sets
$U\subseteq\mathbb{R}_{+}^{l}\times I_{\epsilon_{1}}$ and
$V\subseteq{\mathbf{M}}_{\mathbb{R}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$, such that the
map $\left.G^{(1)}\right|_{U}:U\rightarrow V$ is a homeomorphism.

Let

$\displaystyle
r_{0}=\dfrac{1}{2}\min_{\begin{subarray}{c}\mu,\lambda\in\mathbf{Spec}(x_{0}x_{0}^{*})\\\
\mu\not=\lambda\end{subarray}}|\sqrt{\mu}-\sqrt{\lambda}|.$

For $\gamma_{0}\in(0,r_{0})$ that is small enough, by Lemma 4.2, there exists
$\delta_{0}$ such that for each $x\in{\mathbf{M}}_{\mathbb{R}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$, it has the
singular value decomposition $x=Q_{L}EQ_{R}^{*}$ with
$Q_{L}\in{\mathbf{O}}(d_{1})\cap\mathfrak{B}_{\gamma_{0}/2}\left(\Gamma_{L}^{(1)}(v_{0})\right)$,
$Q_{R}\in{\mathbf{O}}(d_{2})\cap\mathfrak{B}_{\gamma_{0}/2}\left(\Gamma_{R}^{(1)}(v_{0})\right)$
and $E\in{\mathbf{D}}(d)\cap\mathfrak{B}_{\gamma_{0}}(D)$, such that
$|\mathbf{Spec}(xx^{*})|=l$ and $0\notin\mathbf{Spec}(xx^{*})$. Then by the
definition of $r_{0}$, we have

$\displaystyle
E_{1,1}<E_{2,2}<\cdots<E_{L_{1}-1,L_{1}-1}<E_{L_{1},L_{1}}=\cdots=E_{L_{2}-1,L_{2}-1}<E_{L_{2},L_{2}}=\cdots<E_{L_{r},L_{r}}=\cdots=E_{d,d}.$

Denote
$u=(E_{1,1},\dots,E_{L_{1}-1,L_{1}-1},E_{L_{1},L_{1}},E_{L_{2},L_{2}},\ldots,E_{L_{r},L_{r}})\in\mathbb{R}_{+}^{l}$.
Recall that $\pi$ is continuous and $\phi^{(1)}$ is a diffeomorphism. Since
$(Q_{L},Q_{R})\in\left({\mathbf{O}}(d_{1})\times{\mathbf{O}}(d_{2})\right)\cap\mathfrak{B}_{\gamma_{0}}\left(\left(\Gamma_{L}^{(1)}(v_{0}),\Gamma_{R}^{(1)}(v_{0})\right)\right)$
with sufficiently small $\gamma_{0}$, and
$\pi\left(\Gamma_{L}^{(1)}(v_{0}),\Gamma_{R}^{(1)}(v_{0})\right)=\phi^{(1)}(v_{0})$,
we have
$\pi(Q_{L},Q_{R})\in\phi^{(1)}\left(I_{\left(\epsilon_{1}+\|v_{0}\|\right)/2}\right)$.
Thus, we choose $v=\left(\phi^{(1)}\right)^{-1}(\pi(Q_{L},Q_{R}))\in
I_{(\epsilon_{1}+\|v_{0}\|)/2}$, then
$\left(\Gamma_{L}^{(1)}(v),\Gamma_{R}^{(1)}(v)\right)=\sigma_{\pi}^{(1)}\left(\pi(Q_{L},Q_{R})\right)$
is in the same orbit space as $(Q_{L},Q_{R})$ with respect to the right action
of the Lie group $O(d_{2};l_{1},\ldots,l_{r},l_{0})$. Hence, there exists
$g=\mathrm{diag}(g_{d_{1}},g_{l_{0}})\in O(d_{2};l_{1},\ldots,l_{r},l_{0})$
such that $\sigma_{\pi}^{(1)}\circ\pi(Q_{L},Q_{R})=(Q_{L},Q_{R})\cdot
g=(Q_{L}\cdot g_{d_{1}},Q_{R}\cdot g)$. Therefore,

$\displaystyle x=Q_{L}EQ_{R}^{*}=Q_{L}(g_{d_{1}}Eg^{*})Q_{R}^{*}=(Q_{L}\cdot
g_{d_{1}})E(Q_{R}\cdot
g)^{*}=\Gamma_{L}^{(1)}(v)\Delta(u)\Gamma_{R}^{(1)}(v)=G^{(1)}(u,v).$

Now we choose $V={\mathbf{M}}_{\mathbb{R}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$, then $V\subset
G^{(1)}\left(\mathbb{R}_{+}^{l}\times I_{\epsilon_{1}}\right)$. We choose
$U=\left(G^{(1)}\right)^{-1}(V)$, then $\left.G^{(1)}\right|_{U}$ is
surjective. The continuity of the mapping $G^{(1)}$ implies that $U$ is open
in $\mathbb{R}^{l}\times I_{\epsilon_{1}}$. To show $\left.G^{(1)}\right|_{U}$
is a homeomorphism, it suffices to show the following conditions are
satisfied:

  1. (a1)

$\left.G^{(1)}\right|_{U}$ is injective;

  2. (a2)

$\left(\left.G^{(1)}\right|_{U}\right)^{-1}$ is continuous over $V$.

To verify (a1), we assume that $x\in{\mathbf{M}}_{\mathbb{R}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})\cap\mathfrak{B}_{\delta_{0}}(x_{0})$ has the
following singular value decompositions,

$\displaystyle
x=\Gamma_{L}^{(1)}(v)\Delta(u)\Gamma_{R}^{(1)}(v)^{*}=\Gamma_{L}^{(1)}(v^{\prime})\Delta(u^{\prime})\Gamma_{R}^{(1)}(v^{\prime})^{*}.$

If $u\not=u^{\prime}$, we may assume that the $i$th entries of $u$ and
$u^{\prime}$ are different, then the $i$th columns of $\Gamma_{L}^{(1)}(v)$
and $\Gamma_{L}^{(1)}(v^{\prime})$ belong to different eigenspaces of
$xx^{*}$. Following the proof of Lemma 3.3, we will get a contradiction, which
implies that $u=u^{\prime}$. We can also deduce that $v=v^{\prime}$ as in the
proof of Lemma 3.3.

To verify (a2), Consider any sequence $\\{x_{n}\\}_{n\in\mathbb{N}}\subseteq
V$, such that $\lim_{n\rightarrow\infty}x_{n}=x\in V$. Let
$(u_{n},v_{n})=\left(\left.G^{(1)}\right|_{U}\right)^{-1}(x_{n})\in\mathbb{R}_{+}^{l}\times
I_{\epsilon_{1}}$ and
$(u,v)=\left(\left.G^{(1)}\right|_{U}\right)^{-1}(x)\in\mathbb{R}_{+}^{l}\times
I_{\epsilon_{1}}$, then

$\displaystyle
x_{n}=\Gamma_{L}^{(1)}(v_{n})\Delta(u_{n})\Gamma_{R}^{(1)}(v_{n})^{*},\quad
x=\Gamma_{L}^{(1)}(v)\Delta(u)\Gamma_{R}^{(1)}(v)^{*}.$

As in the proof of Lemma 3.3, we can deduce that

$\displaystyle\lim_{n\rightarrow\infty}u_{n}=u.$

Note that $v_{n}\in\overline{I_{(\epsilon_{1}+\|v_{0}\|)/2}}$, which is
compact. Let $v^{\prime}\in\overline{I_{(\epsilon_{1}+\|v_{0}\|)/2}}\subset
I_{\epsilon_{1}}$ be a limit point of the sequence
$\\{v_{n}\\}_{n\in\mathbb{N}}$, then there exists a subsequence
$\\{v_{m_{n}}\\}_{n\in\mathbb{N}}$, such that
$\lim_{n\rightarrow\infty}v_{m_{n}}=v^{\prime}$. By the continuity of the
mappings $\Gamma_{L}^{(1)}$, $\Gamma_{R}^{(1)}$ and $\Delta$, we have

$\displaystyle
x=\lim_{n\rightarrow\infty}x_{m_{n}}=\lim_{n\rightarrow\infty}\Gamma_{L}^{(1)}(v_{m_{n}})\Delta(u_{m_{n}})\Gamma_{R}^{(1)}(v_{m_{n}})^{*}=\Gamma_{L}^{(1)}(v^{\prime})\Delta(u)\Gamma_{R}^{(1)}(v^{\prime})^{*}.$

Thus, the columns of $\Gamma_{L}^{(1)}(v)$ and $\Gamma_{L}^{(1)}(v^{\prime})$
are the eigenvectors of the same eigenvalues of the matrix $xx^{*}$, which
implies that there exists $g_{d_{1}}\in\mathbf{O}(d_{1};l_{1},\ldots,l_{r})$,
such that $\Gamma_{L}^{(1)}(v^{\prime})=\Gamma_{L}^{(1)}(v)g_{d_{1}}$.
Similarly, by considering $x^{*}x$, one can deduce that there exists
$g\in\mathbf{O}(d_{2};l_{1},\ldots,l_{r},l_{0})$, such that
$\Gamma_{R}^{(1)}(v^{\prime})=\Gamma_{R}^{(1)}(v)g$. Hence, we have

$\displaystyle\Delta(u)=\Gamma_{L}^{(1)}(v^{\prime})^{*}x\Gamma_{R}^{(1)}(v^{\prime})=g_{d_{1}}^{*}\Gamma_{L}^{(1)}(v)^{*}x\Gamma_{R}^{(1)}(v)g=g_{d_{1}}^{*}\Delta(u)g.$

By the definition of $\Delta$ and the assumption that
$0\notin\mathbf{Spec}(x_{0}x_{0}^{*})$, if we write
$g=\mathrm{diag}(g_{d_{1}}^{\prime},g_{l_{0}})$, then we have
$g_{d_{1}}^{\prime}=g_{d_{1}}$, which implies that
$\left(\Gamma_{L}^{(1)}(v^{\prime}),\Gamma_{R}^{(1)}(v^{\prime})\right)$ and
$\left(\Gamma_{L}^{(1)}(v),\Gamma_{R}^{(1)}(v)\right)$ are in the same orbit
space of $\pi$, i.e.
$\pi\left(\Gamma_{L}^{(1)}(v),\Gamma_{R}^{(1)}(v)\right)=\pi\left(\Gamma_{L}^{(1)}(v^{\prime}),\Gamma_{R}^{(1)}(v^{\prime})\right)$.
Recall that $\pi\circ\sigma_{\pi}^{(1)}$ is identity and $\phi^{(1)}$ is
diffeomorphism on $I_{\epsilon_{1}}$, we have $v=v^{\prime}$, which implies
that

$\lim_{n\to\infty}v_{n}=v,$

since any limit point $v^{\prime}$ has to be $v$. Thus, we obtain the
continuity of the map $\left(G^{(1)}\right)^{-1}$ and (a2) is proved. The
proof is concluded. ∎

Now we are ready to prove Theorem 1.3.

###### Proof of Theorem 1.3.

The proof is similar to that of Theorem 1.1, which is sketched below.

We first prove Part (i). By Lemma 4.1, there exists a smooth map

$\displaystyle G:\mathbb{R}_{\geq
0}^{l}\times\mathbb{R}^{\frac{1}{2}[d_{2}(d_{2}-1)+d_{1}(d_{1}-1)-\sum_{j=0}^{r}l_{j}(l_{j}-1)]}\rightarrow\mathbb{R}^{d_{1}\times
d_{2}},$

such that ${\mathbf{M}}_{\mathbb{R}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})\subseteq\mathrm{Im}(G)$. In view of Lemma 4.3, we
obtain

$\displaystyle\dim_{{}_{\rm
H}}\left(\mathrm{Im}(G)\right)=d_{1}d_{2}-\frac{1}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2).$
(4.10)

If

$\displaystyle\sum_{j=1}^{N}\frac{1}{H_{j}}<\frac{1}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2),$

then by Lemma 2.3, the translation invariance of Hausdorff measure, (4.10),
Corollary 2.1 and Lemma 2.2, we have

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|s_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d_{1}]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right)$
$\displaystyle=\mathbb{P}\left(Z^{\beta}(t)\in{\mathbf{M}}_{\mathbb{R}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})\mathrm{\ for\ some\ }t\in I\right)$
$\displaystyle\leq\mathbb{P}\left(W^{\beta}(I)\cap
T_{\beta}^{-1}\left(\mathrm{Im}(G)-B^{\beta}\right)\tilde{T}_{\beta}^{-1}\not=\emptyset\right)$
$\displaystyle\leq
c_{6}{\mathbf{H}}_{d_{1}d_{2}-\sum_{j=1}^{N}\frac{1}{H_{j}}}\left(T_{\beta}^{-1}\left(\mathrm{Im}(G)-B^{\beta}\right)\tilde{T}_{\beta}^{-1}\right)=0.$

The remaining case is

$\displaystyle\sum_{j=1}^{N}\frac{1}{H_{j}}=\frac{1}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2).$

To handle this case, for $K\geq 1$, we choose a compact interval
$F_{K}=[0,K]^{l}\times F^{\prime}$, where $F^{\prime}$ is a compact interval
of
$\mathbb{R}^{\frac{1}{2}[d_{2}(d_{2}-1)+d_{1}(d_{1}-1)-\sum_{j=0}^{r}l_{j}(l_{j}-1)]}$
containing the support of $\Gamma_{L}$ and $\Gamma_{R}$ given in Lemma 4.1.
Then by Lemma 2.5, we have

$\displaystyle\lambda_{d_{1}d_{2}}\left(\left(\mathrm{Im}(G|_{F_{K}})-A^{\beta}\right)^{(r)}\right)\leq
Cr^{\frac{1}{2}\sum_{j=1}^{r}(l_{j}-1)(l_{j}+2)}=Cr^{\sum_{j=1}^{N}\frac{1}{H_{j}}},$

for all $r>0$ small, and $C$ is a constant the depends only on
$G,d,H_{1},\ldots,H_{N}$. Here, $G|_{G_{K}}$ is the smooth restriction of $G$
on $F_{K}$, that is $G|_{F_{K}}$ coincide with $G$ on $F_{K}$ and
$G|_{F_{K+1}^{\complement}}=0$. By a similar argument as (3.22), for $K$
large, we obtain

$\displaystyle\mathbb{P}\bigg{(}\left\\{\exists t\in I,\ \mathrm{s.t.\
}\left|s_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d_{1}]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right\\}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\cap\left\\{\mathbf{Spec}\left(Z^{\beta}\left(Z^{\beta}\right)^{*}\right)\subset\left[0,K^{2}\right]\right\\}\bigg{)}$
$\displaystyle\leq\mathbb{P}\left(\left\\{W^{\beta}(t)\in
T_{\beta}^{-1}\left(\mathrm{Im}(G)-B^{\beta}\right)\tilde{T}_{\beta}^{-1}\mathrm{\
for\ some\ }t\in
I\right\\}\cap\left\\{\mathbf{Spec}\left(Z^{\beta}\left(Z^{\beta}\right)^{*}\right)\subset\left[0,K^{2}\right]\right\\}\right)$
$\displaystyle\leq\mathbb{P}\left(W^{\beta}(I)\cap
T_{\beta}^{-1}\left(\mathrm{Im}(G|_{F_{K}})-B^{\beta}\right)\tilde{T}_{\beta}^{-1}\not=\emptyset\right).$

We deduce from Lemma 2.4 and Corollary 2.2 that

$\displaystyle\mathbb{P}\bigg{(}\left\\{\exists t\in I,\ \mathrm{s.t.\
}\left|s_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d_{1}]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq r\right\\}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\cap\left\\{\mathbf{Spec}\left(Z^{\beta}\left(Z^{\beta}\right)^{*}\right)\subset\left[0,K^{2}\right]\right\\}\bigg{)}=0.$

Letting $K\to\infty$, we obtain

$\displaystyle\mathbb{P}\left(\exists t\in I,\ \mathrm{s.t.\
}\left|s_{J_{j}}^{\beta}(t)\right|=1\ \mathrm{for\ some\ disjoint\
}J_{j}\in[d_{1}]\mathrm{\ with\ }|J_{j}|=l_{j},\forall 1\leq j\leq
r\right)=0.$

Next, we prove (ii). We choose $x_{0}\in{\mathbf{M}}_{\mathbb{R}}(d_{1}\times
d_{2};l_{1},\ldots,l_{r})$ satisfying $0\notin\mathbf{Spec}(x_{0}x_{0}^{*})$
and $|\mathbf{Spec}(x_{0}x_{0}^{*})|=l$. By Lemma 4.3, there exists