# Simple matrix models for the flag, Grassmann, and Stiefel manifolds

Lek-Heng Lim  Computational and Applied Mathematics Initiative, Department of
Statistics, University of Chicago, Chicago, IL 60637-1514.
<EMAIL_ADDRESS>and  Ke Ye  KLMM, Academy of Mathematics and Systems
Science, Chinese Academy of Sciences, Beijing 100190, China<EMAIL_ADDRESS>
###### Abstract.

We derive three families of orthogonally-equivariant matrix submanifold models
for the Grassmann, flag, and Stiefel manifolds respectively. These families
are exhaustive — every orthogonally-equivariant submanifold model of the
lowest dimension for any of these manifolds is necessarily a member of the
respective family, with a small number of exceptions. They have several
computationally desirable features. The orthogonal equivariance allows one to
obtain, for various differential geometric objects and operations, closed-form
analytic expressions that are readily computable with standard numerical
linear algebra. The minimal dimension aspect translates directly to a speed
advantage in computations. And having an exhaustive list of all possible
matrix models permits one to identify the model with the lowest matrix
condition number, which translates to an accuracy advantage in computations.
As an interesting aside, we will see that the family of models for the Stiefel
manifold is naturally parameterized by the Cartan manifold, i.e., the positive
definite cone equipped with its natural Riemannian metric.

##  1\. Introduction

As abstract manifolds, the Grassmannian is a set of subspaces, the flag
manifold a set of nested sequences of subspaces, and the Stiefel manifold a
set of frames. To work with these manifolds, not least performing
computations, one needs a model, i.e., a system of extrinsic coordinates, for
them. From the perspective of numerical computations, the best models are
matrix models, representing these manifolds by a quotient manifold or
submanifold of matrices, thereby permitting differential geometric objects to
be expressed in terms of matrix operations computable with standard numerical
linear algebra. Among such models, submanifold models are preferable to
quotient manifold models as points and tangent vectors are represented by
actual matrices instead of equivalence classes of matrices, which require
artificial choices and additional computations to further represent as actual
matrices, bearing in mind that standard numerical linear algebra only works
with actual matrices, not equivalence classes of matrices. For numerical
stability, these models should be equivariant under orthogonal group action,
as orthogonal matrices are the basis of numerically stable algorithms.

For the Grassmannian and flag manifold, these considerations lead us to the
following models. Let $\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n})$ be the
space of $n\times n$ real symmetric matrices, $a,b\in\mathbb{R}$ distinct and
$a_{1},\dots,a_{p+1}\in\mathbb{R}$ generic real numbers. We will show that the
_quadratic model_

$\operatorname{Gr}_{a,b}(k,n)\coloneqq\bigl{\\{}X\in\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n}):(X-aI)(X-bI)=0,\;\operatorname{tr}(X)=ak+b(n-k)\bigr{\\}}$

is diffeomorphic to $\operatorname{Gr}(k,\mathbb{R}^{n})$, the Grassmannian of
$k$-planes in $\mathbb{R}^{n}$; and the _isospectral model_

$\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)\coloneqq\biggl{\\{}X\in\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n}):\prod_{j=0}^{p}(X-a_{j+1}I_{n})=0,\;\operatorname{tr}(X)=\sum_{j=0}^{p}(k_{j+1}-k_{j})a_{j+1}\biggr{\\}}$

is diffeomorphic to $\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n})$,
the manifold of $(k_{1},\dots,k_{p})$-flags in $\mathbb{R}^{n}$. Here we have
assumed that $0<k<n$ and $0\eqqcolon k_{0}<k_{1}<\dots<k_{p+1}\coloneqq n$ are
all integers. Evidently the quadratic model is the $p=1$ case of the
isospectral model.

The quadratic model is so-called because the roots of a monic quadratic matrix
polynomial $(X-aI)(X-bI)=0$ are called quadratic matrices [11], well-known in
studies of numerical range [8, 26]. The special cases $(a,b)=(1,0)$ gives
projection matrices with $X^{2}=X$ and $(a,b)=(1,-1)$ gives involutory
matrices with $X^{2}=I$. So the classical projection model
$\operatorname{Gr}_{1,0}(k,n)=\\{X\in\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n}):X^{2}=X,\;\operatorname{tr}(X)=k\\}$
[19, 23] and the more recent involution model
$\operatorname{Gr}_{1,-1}(k,n)=\\{X\in\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n}):X^{2}=I,\;\operatorname{tr}(X)=2k-n\\}$
[14] are both special cases of the quadratic model.

The descriptions of these models above are implicit. We will prove in Section
4 that they are equivalent to the following explicit descriptions:

$\displaystyle\operatorname{Gr}_{a,b}(k,n)$
$\displaystyle=\biggl{\\{}Q\begin{bmatrix}aI_{k}&0\\\
0&bI_{n-k}\end{bmatrix}Q^{\scriptscriptstyle\mathsf{T}}\in\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n}):Q\in\operatorname{SO}_{n}(\mathbb{R})\biggr{\\}},$
$\displaystyle\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)$
$\displaystyle=\left\\{Q{\begin{bmatrix}a_{1}I_{n_{1}}&0&\cdots&0\\\\[-4.30554pt]
0&a_{2}I_{n_{2}}&&\vdots\\\\[-4.30554pt] \vdots&&\ddots&0\\\
0&\cdots&0&a_{p+1}I_{n_{p+1}}\end{bmatrix}}Q^{\scriptscriptstyle\mathsf{T}}\in\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n}):Q\in\operatorname{SO}_{n}(\mathbb{R})\right\\},$

where $\operatorname{SO}_{n}(\mathbb{R})$ is the set of orthogonal matrices
with unit determinant. This matrix manifold has been in existence for more
than thirty years [5, 9, 16, 1, 12] and is known by the names _isospecral
manifold_ or _spectral manifold_. What is not known is that it is just a
parameterization of the flag manifold, which is of course classical and known
for nearly 120 years since [10] or, in its modern form, for at least fifty
years since [3, Section 5]. To the best of our knowledge, the connection that
they are one and the same has never been made, a minor side contribution of
this article.

One might also ask how these discussions apply to the Stiefel manifold [24] of
orthonormal $k$-frames in $n$-space. As an addendum, we will show in Section 6
that for the Stiefel manifold, there is an analogous family of minimal
$\operatorname{SO}_{n}(\mathbb{R})$-equivariant models, the _Cholesky models_

$\operatorname{V}_{\\!A}(k,n)\coloneqq\\{X\in\mathbb{R}^{n\times
k}:X^{\scriptscriptstyle\mathsf{T}}X=A\\},$

parameterized by
$A\in\operatorname{\mathsf{S}}^{2}_{\scriptscriptstyle++}(\mathbb{R}^{k})$,
the set of $k\times k$ symmetric positive definite matrices. This description
is implicit; but Cholesky models too have an explicit description that
justifies the name,

$\operatorname{V}_{\\!A}(k,n)=\biggl{\\{}Q\begin{bmatrix}R\\\
0\end{bmatrix}:Q\in\operatorname{O}_{n}(\mathbb{R})\biggr{\\}},$

where $R\in\operatorname{GL}_{k}(\mathbb{R})$ is the unique Cholesky factor
with positive diagonal of $A$.

It will become clear in Section 6 that the relevant structure on
$\operatorname{\mathsf{S}}^{2}_{\scriptscriptstyle++}(\mathbb{R}^{k})$ is that
of a Riemannian manifold with metric $\operatorname{tr}(A^{-1}XA^{-1}Y)$, and
not its more common structure as an Euclidean cone. Note that $A=I$ gives the
usual model of the Stiefel manifold as $n\times k$ matrices with orthonormal
columns but more generally a Cholesky model allows for $A^{-1}$-orthonormal
columns.111Fortuitously Stiefel also pioneered the use of
$A^{-1}$-orthonormality in computational mathematics through his conjugate
gradient method [13].

###  1.1. Computational significance

From a computational perspective, these families of models have two desirable
features and are unique in this regard:

  1. (i)

orthogonal equivariance: Let $Q\in\operatorname{O}(n)$. Then
$X\in\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n})$ is in an isospectral model
iff $Q^{\scriptscriptstyle\mathsf{T}}XQ$ is in the model;
$X\in\mathbb{R}^{n\times k}$ is in a Cholesky model iff $QX$ is in the model.

  2. (ii)

minimal dimension: There is no model for the flag (resp. Stiefel) manifold in
an ambient space of dimension smaller than $\frac{1}{2}(n-1)(n+2)$ (resp.
$nk$) with property (i).

  3. (iii)

exhaustive: Any model for the flag (resp. Stiefel) manifold with properties
(i) and (ii) must be among the family of isospectral (resp. Cholesky) models.

Property (i) is key to deriving closed-form analytic expressions for
differential geometric quantities in terms of standard matrix operations
stably computable with numerical linear algebra [14].

Property (ii) ensures that points on the manifold are represented with
matrices of the lowest possible dimension, which is important as the _speed_
of every algorithm in numerical linear algebra depends on the dimension of the
matrices. The current lowest dimensional matrix model of a flag manifold is
obtained by embedding $\operatorname{Flag}(k_{1},\dots,k_{p};\mathbb{R}^{n})$
into a product of Grassmannians as in (9) on p. 9. Even if we use the lowest
dimensional models for these Grassmannians, the result would involve matrices
of dimension $np\times np$ whereas the isospectral model uses only $n\times n$
matrices.

For _accuracy_ , the matrix condition number plays a role analogous to matrix
dimension for speed. Property (iii) allows us to pick from among the
respective family of models the one with the smallest condition number. Every
matrix in an isospectral model
$\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)$ has identical
eigenvalues and therefore condition number $\max(\lvert
a_{1}\rvert,\dots,\lvert a_{p+1}\rvert)/\min(\lvert a_{1}\rvert,\dots,\lvert
a_{p+1}\rvert)$. For the Grassmannian $p=1$ and there is a unique (up to a
constant multiple) perfectly conditioned model with $a=1$, $b=-1$, which is
precisely the involution model in [14]. For more general flag manifolds with
$p>1$, the condition number can be made arbitrarily close to one. For the
Stiefel manifold, the usual model with $A=I$ plays the role of the involution,
namely, as the unique (up to a constant multiple) perfectly conditioned model
among all Cholesky models.

###  1.2. Contributions

The main intellectual effort of this article is to establish property (i) and
half of property (ii) by deriving the isospectral model (Theorem 4.1,
Corollary 4.2) and demonstrating that we may choose $a_{1},\dots,a_{p+1}$ so
that we get a submanifold of
$\operatorname{\mathsf{S}}^{2}_{\circ}(\mathbb{R}^{n})$, the space of
traceless symmetric matrices with
$\dim\operatorname{\mathsf{S}}^{2}_{\circ}(\mathbb{R}^{n})=\frac{1}{2}(n-1)(n+2)$
(Corollary 4.4). To demonstrate the other half of property (ii), i.e., no
lower dimensional model exists, and to prove property (iii), one needs an
amount of representation theory far beyond the scope of our article and is
relegated to [17].

The orthogonal equivariance in property (i) deserves elaboration. Firstly we
really do mean “orthogonal” — every result in this article remains true if
$\operatorname{SO}_{n}(\mathbb{R})$ is replaced by
$\operatorname{O}_{n}(\mathbb{R})$. Secondly we stress the importance of
“equivariance.” The Riemannian metric is often regarded as the center piece of
any computations involving manifolds, not least in manifold optimization. This
is getting things backwards. What is by far more important is equivariance,
or, as a special case, _invariance_. There are uncountably many Riemannian
metrics on the flag manifold even after discounting constant multiples. The
most important Riemannian metrics are the ones that are
$\operatorname{SO}_{n}(\mathbb{R})$-invariant; note that the standard
Euclidean inner product on $\mathbb{R}^{n}$ or $\mathbb{R}^{m\times n}$ has
this property. In the case of the flag, Grassmann, and Stiefel manifolds,
there is an even more special one — the
$\operatorname{SO}_{n}(\mathbb{R})$-invariant Riemannian metric that comes
from the unique bi-invariant metric on $\operatorname{SO}_{n}(\mathbb{R})$. It
is the key to deriving explicit expressions for differential geometric objects
in terms of standard matrix operations readily computable with numerical
linear algebra.

A second goal of this article is show that the Riemannian metrics arising from
our equivariant embeddings of the flag, Grassmann, and Stiefel manifolds are,
up to constant weights, the ones arising from the bi-invariant metric on
$\operatorname{SO}_{n}(\mathbb{R})$ (Section 7). As a perusal of the
computational mathematics literature would reveal, equivariance has never been
brought into center stage. We hope that our article would represent a small
step in this direction.

##  2\. Notations and some background

We generally use blackboard bold fonts for vector spaces and double brackets
$\llbracket\,\cdot\,\rrbracket$ for equivalence classes. On two occasions we
write $\mathbb{Q}_{j}$ for a subspace spanned by columns of an orthogonal
matrix $Q$, which should not cause confusion as the rational field has no role
in this article. We reserve the letter $\mathbb{W}$ for $\mathbb{R}$-vector
spaces and $\mathbb{V}$ for $\operatorname{SO}_{n}(\mathbb{R})$-modules,
usually adorned with various subscripts. We write $\cong$ for diffeomorphisms
of manifolds and $\simeq$ for isomorphisms of vector spaces and modules.

###  2.1. Linear algebra

The real vector spaces of real symmetric, skew-symmetric, and traceless
symmetric matrices will be denoted respectively by

(1) $\displaystyle\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n})$
$\displaystyle\coloneqq\\{X\in\mathbb{R}^{n\times
n}:X^{\scriptscriptstyle\mathsf{T}}=X\\},$
$\displaystyle\operatorname{\mathsf{\Lambda}}^{2}(\mathbb{R}^{n})$
$\displaystyle\coloneqq\\{X\in\mathbb{R}^{n\times
n}:X^{\scriptscriptstyle\mathsf{T}}=-X\\},$
$\displaystyle\operatorname{\mathsf{S}}^{2}_{\circ}(\mathbb{R}^{n})$
$\displaystyle\coloneqq\\{X\in\mathbb{R}^{n\times
n}:X^{\scriptscriptstyle\mathsf{T}}=X,\;\operatorname{tr}(X)=0\\}.$

For the cone of real symmetric positive definite matrices, we write

$\operatorname{\mathsf{S}}^{2}_{\scriptscriptstyle++}(\mathbb{R}^{n})\coloneqq\\{X\in\mathbb{R}^{n\times
n}:y^{\scriptscriptstyle\mathsf{T}}Xy>0\text{ for all }y\neq 0\\}.$

The Lie groups of real orthogonal, special orthogonal, and general linear
groups will be denoted respectively by

(2) $\displaystyle\operatorname{O}_{n}(\mathbb{R})$
$\displaystyle\coloneqq\\{X\in\mathbb{R}^{n\times
n}:X^{\scriptscriptstyle\mathsf{T}}X=I\\},$
$\displaystyle\operatorname{SO}_{n}(\mathbb{R})$
$\displaystyle\coloneqq\\{X\in\mathbb{R}^{n\times
n}:X^{\scriptscriptstyle\mathsf{T}}X=I,\;\det(X)=1\\},$
$\displaystyle\operatorname{GL}_{n}(\mathbb{R})$
$\displaystyle\coloneqq\\{X\in\mathbb{R}^{n\times n}:\det(X)\neq 0\\}.$

Let $n_{1}+\dots+n_{p}=n$. We will also write

$\operatorname{S}(\operatorname{O}_{n_{1}}(\mathbb{R})\times\cdots\times\operatorname{O}_{n_{p}}(\mathbb{R}))\coloneqq\\{\operatorname{diag}(X_{1},\dots,X_{p})\in\operatorname{O}_{n}(\mathbb{R}):\\\
X_{1}\in\operatorname{O}_{n_{1}}(\mathbb{R}),\dots,X_{p}\in\operatorname{O}_{n_{p}}(\mathbb{R}),\;\det(X_{1})\cdots\det(X_{p})=1\\}.$

Note that $\operatorname{SO}_{n}(\mathbb{R})$ is a special case of this. For
each increasing sequence $0\eqqcolon k_{0}<k_{1}<\cdots<k_{p}<k_{p+1}\coloneqq
n$, we also define the _parabolic subgroup_ of block upper triangular
matrices:

$\operatorname{P}_{k_{1},\dots,k_{p}}(\mathbb{R})=\Biggl{\\{}\begin{bmatrix}X_{1,1}&\cdots&X_{1,p+1}\\\
\vdots&\ddots&\vdots\\\ 0&\cdots&X_{p+1,p+1}\\\
\end{bmatrix}\in\operatorname{GL}_{n}(\mathbb{R}):\begin{multlined}X_{ij}\in\mathbb{R}^{(k_{i}-k_{i-1})\times(k_{j}-k_{j-1})},\\\
i,j=1,\dots,p+1\end{multlined}X_{ij}\in\mathbb{R}^{(k_{i}-k_{i-1})\times(k_{j}-k_{j-1})},\\\
i,j=1,\dots,p+1\Biggr{\\}}.$

Let $G$ be a group and $\mathbb{V}$ a vector space. We say that $\mathbb{V}$
is a _$G$ -module_ if there is a _linear_ group action
$G\times\mathbb{V}\to\mathbb{V}$, $(g,v)\mapsto g\cdot v$, i.e., satisfying

$g\cdot(a_{1}v_{1}+a_{2}v_{2})=a_{1}g\cdot v_{1}+a_{2}g\cdot v_{2}$

for any $g\in G$, $a_{1},a_{2}\in\mathbb{R}$, and $v_{1},v_{2}\in\mathbb{V}$.
In this paper, we will mostly limit ourselves to
$G=\operatorname{SO}_{n}(\mathbb{R})$ and two group actions on
$\mathbb{R}^{n}$ and $\mathbb{R}^{n\times n}$ respectively:

(3) $\displaystyle\operatorname{SO}_{n}(\mathbb{R})\times\mathbb{R}^{n}$
$\displaystyle\to\mathbb{R}^{n},\quad$ $\displaystyle(Q,v)$
$\displaystyle\mapsto Q\cdot v\coloneqq Qv,$ (4)
$\displaystyle\operatorname{SO}_{n}(\mathbb{R})\times\mathbb{R}^{n\times n}$
$\displaystyle\to\mathbb{R}^{n\times n},\quad$ $\displaystyle(Q,X)$
$\displaystyle\mapsto Q\cdot X\coloneqq QXQ^{\scriptscriptstyle\mathsf{T}}.$

All matrix subspaces in (1) are also
$\operatorname{SO}_{n}(\mathbb{R})$-modules under the conjugation action (4).
If $\mathbb{V}$ is a $G$-module, then a direct sum of multiple copies
$\mathbb{V}\oplus\dots\oplus\mathbb{V}$ is automatically a $G$-module with
action $Q\cdot(v_{1},\dots,v_{k})\coloneqq(Q\cdot v_{1},\dots,Q\cdot v_{k})$.
So $\mathbb{R}^{n\times k}=\mathbb{R}^{n}\oplus\dots\oplus\mathbb{R}^{n}$ is
also an $\operatorname{SO}_{n}(\mathbb{R})$-module under the action (3).

###  2.2. Differential geometry

We write $\operatorname{Gr}(k,\mathbb{R}^{n})$,
$\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n})$, and
$\operatorname{V}(k,\mathbb{R}^{n})$ respectively for the flag, Grassmann, and
Stiefel manifold as _abstract manifolds_. An element of
$\operatorname{Gr}(k,\mathbb{R}^{n})$ is a subspace
$\mathbb{W}\subseteq\mathbb{R}^{n}$, $\dim\mathbb{W}=k$. An element of
$\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n})$ is a flag, i.e., a
nested sequence of subspaces
$\mathbb{W}_{1}\subseteq\dots\subseteq\mathbb{W}_{p}\subseteq\mathbb{R}^{n}$,
$\dim\mathbb{W}_{i}=k_{i}$, $i=1,\dots,p$. An element of
$\operatorname{V}(k,\mathbb{R}^{n})$ is an orthonormal $k$-frame
$(w_{1},\dots,w_{k})$ in $\mathbb{R}^{n}$.

The abstract Grassmann and flag manifolds are _$G$ -manifolds_ for any
subgroup $G\subseteq\operatorname{GL}_{n}(\mathbb{R})$, i.e., they come
naturally equipped with a $G$-action. For Grassmann manifolds, take any
$\mathbb{W}\in\operatorname{Gr}(k,\mathbb{R}^{n})$ and any $X\in
G\subseteq\operatorname{GL}_{n}(\mathbb{R})$, the action is given by

$X\cdot\mathbb{W}\coloneqq\\{Xw\in\mathbb{R}^{n}:w\in\mathbb{W}\\},$

noting that $\dim X\cdot\mathbb{W}=\dim\mathbb{W}$. This action extends to
$\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n})$ since if
$\mathbb{W}_{1}\subseteq\dots\subseteq\mathbb{W}_{p}$, then
$X\cdot\mathbb{W}_{1}\subseteq\dots\subseteq X\cdot\mathbb{W}_{p}$ for any
$X\in G\subseteq\operatorname{GL}_{n}(\mathbb{R})$. So we may write

(5)
$X\cdot(\mathbb{W}_{1}\subseteq\dots\subseteq\mathbb{W}_{p})\coloneqq(X\cdot\mathbb{W}_{1}\subseteq\dots\subseteq
X\cdot\mathbb{W}_{p}),$

again noting that since dimensions are preserved we have a well-defined
action. If $G$ is any of the groups in (2), then this action is transitive.

The abstract Stiefel manifold is a $G$-manifold for any subgroup
$G\subseteq\operatorname{O}_{n}(\mathbb{R})$ via the action

(6) $X\cdot(w_{1},\dots,w_{k})\coloneqq(Xw_{1},\dots,Xw_{k})$

for any $(w_{1},\dots,w_{k})\in\operatorname{V}(k,\mathbb{R}^{n})$. This
action is transitive if $G=\operatorname{O}_{n}(\mathbb{R})$ or
$\operatorname{SO}_{n}(\mathbb{R})$.

A notion central to this article is that of an equivariant embedding, which
has been extensively studied in a more general setting [21, 4, 18, 25].

######  Definition 2.1 (Equivariant embedding and equivariant submanifold).

Let $G$ be a group, $\mathbb{V}$ be a $G$-module, and $\mathcal{M}$ be a
$G$-manifold. An embedding $\varepsilon:\mathcal{M}\to\mathbb{V}$ is called a
$G$-equivariant embedding if $\varepsilon(g\cdot x)=g\cdot\varepsilon(x)$ for
all $x\in\mathcal{M}$ and $g\in G$. In this case, the embedded image
$\varepsilon(\mathcal{M})$ is called a $G$-submanifold of $\mathbb{V}$.

For this article, we only need to know two special cases. For the flag
manifold (and thus Grassmannian when $p=1$),
$G=\operatorname{SO}_{n}(\mathbb{R})$,
$\mathcal{M}=\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n})$,
$\mathbb{V}=\mathbb{R}^{n\times n}$, $G$ acts on $\mathbb{V}$ via (4) and on
$\mathcal{M}$ via (5). For the Stiefel manifold,
$G=\operatorname{SO}_{n}(\mathbb{R})$, $\mathcal{M}=\operatorname{V}(k,n)$,
$\mathbb{V}=\mathbb{R}^{n\times k}$, $G$ acts on $\mathbb{V}$ via (3) and on
$\mathcal{M}$ via (6).

##  3\. Existing models of the flag manifold

The reader may find a list of all known existing matrix models of the
Grassmannian and Stiefel manifold in [15]. Models for the flag manifold are
more obscure and we review a few relevant ones here. The model most commonly
used in pure mathematics is as a quotient manifold,

(7)
$\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n})\cong\operatorname{GL}_{n}(\mathbb{R})/\operatorname{P}_{k_{1},\dots,k_{p}}(\mathbb{R}),$

with the parabolic subgroup as defined in Section 2.1.

As for known222The authors of [5, 9, 16, 1, 12] did not appear to know that
they are discussing the flag manifold. models of the flag manifold in applied
mathematics, the only ones we are aware of were first proposed in [27]. We
will highlight two: There is the orthogonal analogue of (7), as the quotient
manifold

(8)
$\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n})\cong\operatorname{SO}_{n}(\mathbb{R})/\operatorname{S}(\operatorname{O}_{n_{1}}(\mathbb{R})\times\cdots\times\operatorname{O}_{n_{p+1}}(\mathbb{R})).$

It has been shown in [27] that any flag manifold may be embedded as a
submanifold in a product of Grassmannians so that any model for the latter
yields a model for the former. We recall this result here: Let
$n_{1},\dots,n_{p+1}\in\mathbb{N}$ with $n_{1}+\dots+n_{p+1}=n$. Define

(9)
$\operatorname{Flag}_{\operatorname{Gr}}(n_{1},\dots,n_{p+1})\coloneqq\bigl{\\{}(\mathbb{W}_{1},\dots,\mathbb{W}_{p+1})\in\operatorname{Gr}(n_{1},\mathbb{R}^{n})\times\dots\times\operatorname{Gr}(n_{p+1},\mathbb{R}^{n}):\\\
\mathbb{W}_{1}\oplus\dots\oplus\mathbb{W}_{p+1}=\mathbb{R}^{n}\bigr{\\}},$

with $\oplus$ the orthogonal direct sum of subspaces. Then every flag manifold
is diffeomorphic to a submanifold of the form in (9) via the following lemma,
where we have also included (7) and (8) for completeness. Although we use
$\varphi_{1}$ for a different purpose in this article, its existence carries
the important implication that every model of the Grassmannian automatically
gives one for the flag manifold.

######  Lemma 3.1 (Change-of-coordinates for flag manifolds I).

Let $0\eqqcolon k_{0}<k_{1}<\cdots<k_{p}<k_{p+1}\coloneqq n$ be integers and

(10) $n_{i+1}=k_{i+1}-k_{i},\quad i=1,\dots,p.$

Then the maps below are all diffeomorphisms:

${{\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n})}}$${{\operatorname{Flag}_{\operatorname{Gr}}(n_{1},\dots,n_{p+1})}}$${{\operatorname{GL}_{n}(\mathbb{R})/\operatorname{P}_{k_{1},\dots,k_{p}}(\mathbb{R})}}$${{\operatorname{SO}_{n}(\mathbb{R})/\operatorname{S}(\operatorname{O}_{n_{1}}(\mathbb{R})\times\cdots\times\operatorname{O}_{n_{p+1}}(\mathbb{R}))}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$$\scriptstyle{\varphi_{3}}$

with $\varphi_{1},\varphi_{2},\varphi_{3}$ defined by

$\displaystyle\varphi_{1}(\mathbb{W}_{1}\subseteq\cdots\subseteq\mathbb{W}_{p})=(\mathbb{W}_{1},\mathbb{W}_{2}/\mathbb{W}_{1},\dots,\mathbb{W}_{p}/\mathbb{W}_{p-1},\mathbb{R}^{n}/\mathbb{W}_{p}),$
$\displaystyle\varphi_{2}(\llbracket
X\rrbracket_{\operatorname{GL}})=(\mathbb{X}_{1}\subseteq\cdots\subseteq\mathbb{X}_{p}),\qquad\varphi_{3}(\llbracket
Q\rrbracket_{\operatorname{SO}})=(\mathbb{Q}_{1},\dots,\mathbb{Q}_{p+1}),$

where for each $i=1,\dots,p$ and $j=1,\dots,p+1$,

  1. (a)

$\mathbb{X}_{i}$ is the vector space spanned by the first $k_{i}$ columns of
$X\in\operatorname{GL}_{n}(\mathbb{R})$;

  2. (b)

$\mathbb{W}_{i+1}/\mathbb{W}_{i}$ is the orthogonal complement of
$\mathbb{W}_{i}$ in $\mathbb{W}_{i+1}$, $\mathbb{W}_{0}\coloneqq\\{0\\}$, and
$\mathbb{W}_{p+1}\coloneqq\mathbb{R}^{n}$;

  3. (c)

$\mathbb{Q}_{j}$ is the vector space spanned by column vectors of
$Q\in\operatorname{SO}_{n}(\mathbb{R})$ in columns $k_{j-1}+1,\dots,k_{j}$.

###### Proof.

It has been shown in [27, Proposition 3] that $\varphi_{1}$ is a
diffeomorphism, leaving $\varphi_{2}$ and $\varphi_{3}$. Let
$e_{1},\dots,e_{n}\in\mathbb{R}^{n}$ be the standard basis vectors and set
$\mathbb{E}_{j}\coloneqq\operatorname{span}_{\mathbb{R}}\\{e_{1},\dots,e_{j}\\}$,
$j=1,\dots,n$. Consider the map

$\rho_{2}:\operatorname{GL}_{n}(\mathbb{R})\to\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n}),\quad
X\mapsto X\cdot(\mathbb{E}_{k_{1}}\subseteq\cdots\subseteq\mathbb{E}_{k_{p}})$

with the action $\cdot$ in (5). This map is surjective as the action of
$\operatorname{GL}_{n}(\mathbb{R})$ is transitive. The stabilizer of
$(\mathbb{E}_{k_{1}}\subseteq\cdots\subseteq\mathbb{E}_{k_{p}})\in\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n})$
is easily seen to be the parabolic subgroup
$P_{k_{1},\dots,k_{p}}(\mathbb{R})$ in Section 2.1. The orbit-stabilizer
theorem then yields the required diffeomorphism $\varphi_{2}$ from $\rho_{2}$.
The same argument appied to

$\rho_{3}:\operatorname{SO}_{n}(\mathbb{R})\to\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n}),\quad
Q\mapsto Q\cdot(\mathbb{E}_{k_{1}}\subseteq\cdots\subseteq\mathbb{E}_{k_{p}})$

yields the required diffeomorphism $\varphi_{3}$. ∎

In Section 5, we will add to the list of diffeomorphisms in Lemma 3.1 after
establishing various properties of the isospectral model.

##  4\. Equivariant matrix models of the Grassmannian and flag manifold

We begin by deriving the isospectral model
$\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)$, showing that
any $\operatorname{SO}_{n}(\mathbb{R})$-equivariant embedding of
$\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n})$ into
$\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n})$ must be an isospectral model.
Among these isospectral models are a special class of traceless isospectral
models when we choose $a_{1},\dots,a_{p+1}$ to satisfy

$\sum_{i=0}^{p}a_{i+1}(k_{i+1}-k_{i})=0.$

We have shown in [17] that very
$\operatorname{SO}_{n}(\mathbb{R})$-equivariant model of a flag manifold that
is _minimal_ , i.e., whose ambient space has the smallest possible dimension,
must necessarily be a traceless isospectral model. In other words, no matter
what space $\mathbb{V}$ we embed
$\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n})$ in, as long as the
embedding $\varepsilon$ is equivariant and the dimension of $\mathbb{V}$ is
the smallest, then (a) we must have
$\mathbb{V}\simeq\operatorname{\mathsf{S}}^{2}_{\circ}(\mathbb{R}^{n})$ and
(b) the image of $\varepsilon$ must be a traceless isospectral model. The
proof of (a) requires a heavy does of representation theory beyond the scope
of this article but we will say a few words about it in Corollary 4.3 to
highlight this property for a special case. The claim in (b) follows from
Theorem 4.1 and Corollary 4.3.

The isospectral model given below in (11) appears more complex than the one
presented in Section 1 but it will be simplified later in Proposition 4.5 for
generic parameters.

######  Theorem 4.1 (Isospectral model I).

Let $0\eqqcolon k_{0}<k_{1}<\cdots<k_{p}<k_{p+1}\coloneqq n$ be integers and

$n_{i+1}\coloneqq k_{i+1}-k_{i},\quad i=1,\dots,p.$

If
$\varepsilon:\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n})\to\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n})$
is an $\operatorname{SO}_{n}(\mathbb{R})$-equivariant embedding, then its
image must take the form

(11)
$\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)\coloneqq\biggl{\\{}X\in\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n}):\prod_{j=0}^{p}(X-a_{j+1}I_{n})=0,\\\\[-12.91663pt]
\operatorname{tr}(X^{i})=\sum_{j=0}^{p}n_{j+1}a_{j+1}^{i},\;i=1,\dots,p\biggr{\\}}$

for some distinct $a_{1},\dots,a_{p+1}\in\mathbb{R}$.

###### Proof.

Since $\varepsilon$ is $\operatorname{SO}_{n}(\mathbb{R})$-equivariant, its
image $\varepsilon(\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n}))$ is
the orbit of a point
$B\in\varepsilon(\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n}))$ under
the action of $\operatorname{SO}_{n}(\mathbb{R}^{n})$ on
$\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n})$. Let $b_{1}>\cdots>b_{q+1}$ be
the distinct eigenvalues of $B$ of multiplicities $m_{1},\dots,m_{q+1}$. Then
we may assume that

$B=\begin{bmatrix}b_{1}I_{m_{1}}&\cdots&0\\\ \vdots&\ddots&\vdots\\\
0&\cdots&b_{p}I_{m_{q+1}}\end{bmatrix}.$

It follows from the orbit-stabilizer theorem that

$\displaystyle\operatorname{SO}_{n}(\mathbb{R})/\operatorname{S}(\operatorname{O}_{n_{1}}(\mathbb{R})\times\cdots\times\operatorname{O}_{n_{p+1}}(\mathbb{R}))$
$\displaystyle\simeq\varepsilon(\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n}))$
$\displaystyle\simeq\operatorname{SO}_{n}(\mathbb{R})/\operatorname{S}(\operatorname{O}_{m_{1}}(\mathbb{R})\times\cdots\times\operatorname{O}_{m_{q+1}}(\mathbb{R})),$

from which we deduce that $q=p$ and
$\\{n_{1},\dots,n_{p+1}\\}=\\{m_{1},\dots,m_{q+1}\\}$. Let
$\sigma\in\mathfrak{S}_{p+1}$ be such that
$n_{1}=m_{\sigma(1)},\dots,n_{p+1}=m_{\sigma(p+1)}$. Set $a_{1}\coloneqq
b_{\sigma(1)},\dots,a_{1}\coloneqq b_{\sigma(1)}$. Then
$\varepsilon(\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n}))\subseteq\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)$.
For the reverse inclusion, since $(X-a_{1}I)\cdots(X-a_{p+1}I)=0$, any
$X\in\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)$ has
eigenvalues $a_{1},\dots,a_{p+1}$ with multiplicities $n_{1},\dots,n_{p+1}$
determined by the Vandermonde system

(12) $\begin{bmatrix}1&1&\cdots&1&1\\\ a_{1}&a_{2}&\cdots&a_{p}&a_{p+1}\\\
a^{2}_{1}&a^{2}_{2}&\cdots&a^{2}_{p}&a^{2}_{p+1}\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
a^{p}_{1}&a^{p}_{2}&\cdots&a^{p}_{p}&a^{p}_{p+1}\end{bmatrix}\begin{bmatrix}n_{1}\\\
n_{2}\\\ n_{3}\\\ \vdots\\\ n_{p+1}\end{bmatrix}=\begin{bmatrix}n\\\
\operatorname{tr}(X)\\\ \operatorname{tr}(X^{2})\\\ \vdots\\\
\operatorname{tr}(X^{p})\end{bmatrix}.$

Since we also have that $X\in\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n})$,
it must take the form
$X=Q\operatorname{diag}(a_{1}I_{n_{1}},\dots,a_{p+1}I_{n_{p+1}})Q^{\scriptscriptstyle\mathsf{T}}$
for some $Q\in\operatorname{SO}_{n}(\mathbb{R})$. Hence
$\varepsilon(\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n}))\supseteq\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)$.
∎

Embedded in the proof of Theorem 4.1 is an alternative parametric
characterization of the isospectral model worth stating separately, and which
also shows that the object called “isospectral manifold” in [9] is exactly a
flag manifold.

######  Corollary 4.2 (Isospectral model II).

Let $k_{0},\dots,k_{p+1},n,n_{1},\dots,n_{p+1}$ and $a_{1},\dots,a_{p+1}$ be
as in Theorem 4.1. Then

(13)
$\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)=\left\\{Q{\begin{bmatrix}a_{1}I_{n_{1}}&0&\cdots&0\\\\[-4.30554pt]
0&a_{2}I_{n_{2}}&&\vdots\\\\[-4.30554pt] \vdots&&\ddots&0\\\
0&\cdots&0&a_{p+1}I_{n_{p+1}}\end{bmatrix}}Q^{\scriptscriptstyle\mathsf{T}}\in\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n}):Q\in\operatorname{SO}_{n}(\mathbb{R})\right\\}.$

As we alluded to in Section 1, we call (11) or (13) the _isospectral model_.
There are two special cases worth highlighting separately.

######  Corollary 4.3 (Traceless isospectral model).

Let $k_{0},\dots,k_{p+1},n,n_{1},\dots,n_{p+1}$ be as in Theorem 4.1. Let
$a_{1},\dots,a_{p+1}\in\mathbb{R}$ be such that

(14) $\sum_{j=0}^{p}n_{j+1}a_{j+1}=\sum_{j=0}^{p}(k_{j+1}-k_{j})a_{j+1}=0.$

Then

$\displaystyle\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)$
$\displaystyle=\biggl{\\{}X\in\operatorname{\mathsf{S}}^{2}_{\circ}(\mathbb{R}^{n}):\prod_{j=0}^{p}(X-a_{j+1}I_{n})=0,$
$\displaystyle\hskip
107.63855pt\operatorname{tr}(X^{i})=\sum_{j=0}^{p}n_{j+1}a_{j+1}^{i},\;i=1,\dots,p\biggr{\\}}$
$\displaystyle=\bigl{\\{}Q\operatorname{diag}(a_{1}I_{n_{1}},a_{2}I_{n_{2}},\dots,a_{p+1}I_{n_{p+1}})Q^{\scriptscriptstyle\mathsf{T}}\in\operatorname{\mathsf{S}}^{2}_{\circ}(\mathbb{R}^{n}):Q\in\operatorname{SO}_{n}(\mathbb{R})\bigr{\\}}$

has the lowest possible dimension ambient space among all possible
$\operatorname{SO}_{n}(\mathbb{R})$-equivariant models of the flag manifold
whenever $n\geq 17$.

###### Proof.

Note that the difference here is that the matrices are traceless, i.e., we
have

$\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)\subseteq\operatorname{\mathsf{S}}^{2}_{\circ}(\mathbb{R}^{n})$

and this is obvious from the parametric characterization as any
$X\in\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)$ has
$\operatorname{tr}(X)$ given by the expression in (14). The minimal
dimensionality and exhaustiveness of these traceless isospectral models when
$n\geq 17$ have been established in [17, Theorem 3.5 and Proposition 3.7].
Nevertheless, in part as a demonstration of how such a result is plausible, we
will prove a special case in Proposition 4.7 that avoids group representation
theory entirely. ∎

The $p=1$ special case of Theorem 4.1 is also worth stating separately. It
gives a complete classification of equivariant models of the Grassmannian,
namely, they must all be quadratic models.

######  Corollary 4.4 (Quadratic model).

If
$\varepsilon:\operatorname{Gr}(k,\mathbb{R}^{n})\to\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n})$
is an $\operatorname{SO}_{n}(\mathbb{R})$-equivariant embedding, then its
image must take the form

$\displaystyle\operatorname{Gr}_{a,b}(k,n)$
$\displaystyle\coloneqq\bigl{\\{}X\in\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n}):(X-aI_{n})(X-bI_{n})=0,\;\operatorname{tr}(X)=ak+b(n-k)\bigr{\\}}$
$\displaystyle=\biggl{\\{}Q\begin{bmatrix}aI_{k}&0\\\
0&bI_{n-k}\end{bmatrix}Q^{\scriptscriptstyle\mathsf{T}}\in\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n}):Q\in\operatorname{SO}_{n}(\mathbb{R})\biggr{\\}}$

for some distinct $a,b\in\mathbb{R}$.

A notable aspect of the quadratic model is that as $p=1$, the Vandermonde
system (12) in the defining conditions of (11) reduces to a single trace
condition. It turns out that generically this reduction always holds, even
when $p>1$. In other words, all we need are the first two rows of (12),

$\displaystyle n$ $\displaystyle=n_{1}+\dots+n_{p+1},$
$\displaystyle\operatorname{tr}(X)$
$\displaystyle=a_{1}n_{1}+\dots+a_{p+1}n_{p+1}.$

The reason is that $n_{1},\dots,n_{p+1}$ are positive integers, which greatly
limits the number of possible solutions; in fact for generic values, the two
equations above have unique positive integer solutions $n_{1},\dots,n_{p+1}$.
There is no need to look at the remaining rows of (12).

######  Proposition 4.5 (Simpler isospectral model).

Let $k_{0},\dots,k_{p+1},n,n_{1},\dots,n_{p+1}$ be as in Theorem 4.1. For
generic $a_{1},\dots,a_{p+1}\in\mathbb{R}$,

$\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)=\biggl{\\{}X\in\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n}):\prod_{j=0}^{p}(X-a_{j+1}I_{n})=0,\;\operatorname{tr}(X)=\sum_{j=0}^{p}n_{j+1}a_{j+1}\biggr{\\}}.$

###### Proof.

Let $t\coloneqq n_{1}a_{1}+\dots+n_{p+1}a_{p+1}$ and denote the set on the
right by $\operatorname{Fl}_{a_{1},\dots,a_{p+1}}(t,n)$. For any distinct
$a_{1},\dots,a_{p+1}\in\mathbb{R}$,

$\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)\subseteq\operatorname{Fl}_{a_{1},\dots,a_{p+1}}(t,n).$

We will show that the reverse inclusion holds for generic values of
$a_{1},\dots,a_{p+1}$. Let

$\displaystyle M$
$\displaystyle\coloneqq\\{(n_{1},\dots,n_{p+1})\in\mathbb{N}^{p+1}:n_{1}+\cdots+n_{p+1}=n\\},$
$\displaystyle H$
$\displaystyle\coloneqq\\{m-m^{\prime}\in\mathbb{Z}^{p+1}:m\neq
m^{\prime},\;m,m^{\prime}\in M\\},$ (15) $\displaystyle C$
$\displaystyle\coloneqq\\{(a_{1},\dots,a_{p+1})\in\mathbb{R}^{p+1}:a_{i}\neq
a_{j}\text{ if }i\neq j\\}.$

The hyperplane defined by $h\in H$ is

$h^{\perp}\coloneqq\\{a\in\mathbb{R}^{p+1}:a^{\scriptscriptstyle\mathsf{T}}h=a_{1}h_{1}+\dots+a_{p+1}h_{p+1}=0\\}.$

Since $H$ is a finite set and $C$ is a complement of a union of finitely many
hyperplanes, $C\setminus\bigcup_{h\in H}h^{\perp}$ is also a complement of a
union of finitely many hyperplanes. For any $a\in C\setminus\bigcup_{h\in
H}h^{\perp}$ and $m,m^{\prime}\in M$,
$a^{\scriptscriptstyle\mathsf{T}}m=a^{\scriptscriptstyle\mathsf{T}}m^{\prime}$
implies that $m=m^{\prime}$. For any
$X\in\operatorname{Fl}_{a_{1},\dots,a_{p+1}}(t,n)$, the multiplicities
$n_{1},\dots,n_{p+1}$ of $a_{1},\dots,a_{p+1}$ are uniquely determined by the
single equation $\operatorname{tr}(X)=a_{1}n_{1}+\dots+a_{p+1}n_{p+1}$. Hence
$X\in\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)$. ∎

The genericity assumption on $a_{1},\dots,a_{p+1}$ is essential for the
simplified description in Proposition 4.5 as the following example shows.

######  Example 4.6.

Let $n=5$, $p=2$, $(a_{1},a_{2},a_{3})=(0,1,2)$, and consider the isospectral
model $\operatorname{Flag}_{0,1,2}(1,4,5)$. Then $t=5$ and we have
$\operatorname{Flag}_{0,1,2}(1,4,5)\subseteq\operatorname{Fl}_{0,1,2}(5,5)$.
To see that this inclusion is strict, take

$A=\begin{bmatrix}0&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&1&0&0\\\ 0&0&0&1&0\\\
0&0&0&0&2\\\ \end{bmatrix},\quad B=\begin{bmatrix}0&0&0&0&0\\\ 0&0&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&2&0\\\ 0&0&0&0&2\\\ \end{bmatrix},$

and observe that
$A\in\operatorname{Flag}_{0,1,2}(1,4,5)\neq\operatorname{Flag}_{0,1,2}(2,3,5)\ni
B$. We have a disjoint union of nonempty sets

$\operatorname{Flag}_{0,1,2}(1,4,5)\sqcup\operatorname{Flag}_{0,1,2}(2,3,5)=\operatorname{Fl}_{0,1,2}(5,5),$

which implies that

$\operatorname{Flag}_{0,1,2}(1,4,5)\subsetneq\operatorname{Fl}_{0,1,2}(5,5).$

Nevertheless the point of Proposition 4.5 is that there will always be some
other choices, in fact uncountably many, of $a_{1},a_{2},a_{3}$ that work.
Take $(a_{1},a_{2},a_{3})=(0,1,\sqrt{2})$. Then $t=2\sqrt{2}+1$ and

$\displaystyle n_{1}+n_{2}+n_{3}$ $\displaystyle=5,$ $\displaystyle
a_{1}n_{1}+a_{2}n_{2}+a_{3}n_{3}$ $\displaystyle=2\sqrt{2}+1,$

have the unique positive integer solution $(n_{1},n_{2},n_{3})=(2,1,2)$. Hence

$\operatorname{Flag}_{0,1,2}(2,3,5)=\operatorname{Fl}_{0,1,2}(2\sqrt{2}+1,5).$

Special values of $a_{1},\dots,a_{p+1}$ in Theorem 4.1 give us specific models
with various desirable features. For instance, we have mentioned the
involution model [14] that parameterizes the Grassmannian with perfectly
conditioned matrices, obtained by setting $(a,b)=(1,-1)$ in Corollary 4.4.

For $n=2$,
$\operatorname{Flag}(1,\mathbb{R}^{2})=\operatorname{Gr}(1,\mathbb{R}^{2})$
differs from other cases (because $\operatorname{SO}_{2}(\mathbb{R})$ is
abelian unlike $\operatorname{SO}_{n}(\mathbb{R})$ for $n\geq 3$) and has to
be treated separately. In this case all minimal
$\operatorname{SO}_{2}(\mathbb{R})$-equivariant embeddings of
$\operatorname{Gr}(1,\mathbb{R}^{2})$ may be easily characterized.

######  Proposition 4.7 (Minimal equivariant models of
$\operatorname{Flag}(1,\mathbb{R}^{2})=\operatorname{Gr}(1,\mathbb{R}^{2})$).

Any $\operatorname{SO}_{2}(\mathbb{R})$-equivariant embedding of
$\operatorname{Gr}(1,\mathbb{R}^{2})$ into
$\operatorname{\mathsf{S}}^{2}_{\circ}(\mathbb{R}^{2})$ must take the form

$\operatorname{Gr}(1,\mathbb{R}^{2})\to\operatorname{\mathsf{S}}^{2}_{\circ}(\mathbb{R}^{2}),\quad\operatorname{span}\biggl{\\{}\begin{bmatrix}x\\\
y\end{bmatrix}\biggr{\\}}\mapsto\frac{r}{\sqrt{x^{2}+y^{2}}}\begin{bmatrix}cx-
sy&sx+cy\\\ sx+cy&-cx+sy\end{bmatrix}$

for some $r>0$ and some $\begin{bmatrix}c&-s\\\
s&c\end{bmatrix}\in\operatorname{SO}_{2}(\mathbb{R})$. All such embeddings are
minimal.

###### Proof.

Write $\mathbb{S}^{1}\coloneqq\\{\begin{bmatrix}x\\\
y\end{bmatrix}\in\mathbb{R}^{2}:x^{2}+y^{2}=1\\}$ for the unit sphere in
$\mathbb{R}^{2}$. Recall that
$\operatorname{Gr}(1,\mathbb{R}^{2})=\mathbb{RP}^{1}$, the real projective
line. Let
$f:\mathbb{RP}^{1}\to\operatorname{\mathsf{S}}^{2}_{\circ}(\mathbb{R}^{2})$ be
an $\operatorname{SO}_{2}(\mathbb{R})$-equivariant embedding. We consider

(16)
$j:\mathbb{S}^{1}\cong\mathbb{RP}^{1}\xrightarrow{f}\operatorname{\mathsf{S}}^{2}_{\circ}(\mathbb{R}^{2})\cong\mathbb{R}^{2}$

where the first $\cong$ is the usual stereographic projection of
$\mathbb{S}^{1}$ onto $\mathbb{R}$ with north pole mapping to the point-at-
infinity; and the second $\cong$ is
$\operatorname{\mathsf{S}}^{2}_{\circ}(\mathbb{R}^{2})\to\mathbb{R}^{2}$,
$\begin{bmatrix}x&y\\\ y&-x\end{bmatrix}\mapsto\begin{bmatrix}x\\\
y\end{bmatrix}$. It is easy to see that both are
$\operatorname{SO}_{2}(\mathbb{R})$-equivariant. It remains to characterize
all $\operatorname{SO}_{2}(\mathbb{R})$-equivariant embeddings
$j:\mathbb{S}^{1}\to\mathbb{R}^{2}$.

By equivariance, we must have

$j(\mathbb{S}^{1})=\left\\{\begin{bmatrix}c&-s\\\
s&c\end{bmatrix}\begin{bmatrix}x\\\
y\end{bmatrix}\in\mathbb{R}^{2}:\begin{bmatrix}c&-s\\\
s&c\end{bmatrix}\in\operatorname{SO}_{2}(\mathbb{R})\right\\},$

where $j(u_{0})=\begin{bmatrix}x_{0}\\\
y_{0}\end{bmatrix}\in\mathbb{R}^{2}\setminus\\{0\\}$ for a fixed
$u_{0}\in\mathbb{S}^{1}$. Let

$r_{0}=(x_{0}^{2}+y_{0}^{2})^{1/2}>0,\qquad
v_{0}=\frac{1}{\sqrt{x_{0}^{2}+y_{0}^{2}}}\begin{bmatrix}x_{0}\\\
y_{0}\end{bmatrix}\in\mathbb{S}^{1}.$

Then $j(\mathbb{S}^{1})=r_{0}\mathbb{S}^{1}$. Let $Q=\begin{bmatrix}c&-s\\\
s&c\end{bmatrix}\in\operatorname{SO}_{2}(\mathbb{R})$ be such that
$Qu_{0}=v_{0}$. Then
$v_{0}=r_{0}^{-1}j(u_{0})=r_{0}^{-1}j(Q^{\scriptscriptstyle\mathsf{T}}v_{0})$.
So the map $\mathbb{S}^{1}\to\mathbb{R}^{2}$, $v\mapsto
r^{-1}j(Q^{\scriptscriptstyle\mathsf{T}}v)$ is the inclusion
$\mathbb{S}^{1}\subseteq\mathbb{R}^{2}$. Hence we have

$j(v)=r\begin{bmatrix}cx-sy\\\ sx+cy\end{bmatrix}$

for $v=\begin{bmatrix}x\\\ y\end{bmatrix}\in\mathbb{S}^{1}$ and
$Q=\begin{bmatrix}c&-s\\\
s&c\end{bmatrix}\in\operatorname{SO}_{2}(\mathbb{R})$. Composing $j$ with the
inverses of the two diffeomorphisms in (16), we obtain

$f\biggl{(}\operatorname{span}\biggl{\\{}\begin{bmatrix}x\\\
y\end{bmatrix}\biggr{\\}}\biggr{)}=\frac{r}{\sqrt{x^{2}+y^{2}}}\begin{bmatrix}cx-
sy&sx+cy\\\ sx+cy&-cx+sy\end{bmatrix}$

for any $\operatorname{span}\bigl{\\{}\begin{bmatrix}x\\\
y\end{bmatrix}\bigr{\\}}\in\mathbb{RP}^{1}$. If the embedding above is not of
minimal dimension, then there is an embedding of $\mathbb{S}^{1}$ into
$\mathbb{R}^{1}$. The image of $\mathbb{S}^{1}$ in $\mathbb{R}^{1}$ is
connected and compact and thus a closed interval $[a,b]$, a contradiction as
$[a,b]$ is contractible and $\mathbb{S}^{1}$ is not. ∎

##  5\. Change-of-coordinates for isospectral models

In this section we provide change-of-coordinates formulas to transform from
one model of the flag manifold or Grassmannian to another, focusing on the
isospectral and quadratic models. We begin with the following addendum to
Lemma 3.1.

######  Lemma 5.1 (Change-of-coordinates for flag manifold II).

Let $k_{0},\dots,k_{p+1},n,n_{1},\dots,n_{p+1}$ and $a_{1},\dots,a_{p+1}$ be
as in Theorem 4.1. Using the parametric characterization in Corollary 4.2, the
map

$\displaystyle\varphi_{4}:\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)$
$\displaystyle\to\operatorname{Flag}(k_{1},\dots,k_{p};\mathbb{R}^{n}),$
$\displaystyle
Q\operatorname{diag}(a_{1}I_{n_{1}},a_{2}I_{n_{2}},\dots,a_{p+1}I_{n_{p+1}})Q^{\scriptscriptstyle\mathsf{T}}$
$\displaystyle\mapsto(\mathbb{Q}_{1},\dots,\mathbb{Q}_{p}),$

where $\mathbb{Q}_{j}\subseteq\mathbb{R}^{n}$ is the subspace spanned by the
first $k_{j}$ column vectors of $Q$, $j=1,\dots,p$, is a diffeomorphism.

###### Proof.

First we need to check that $\varphi_{4}$ is well-defined. Write

(17)
$\Lambda_{a}\coloneqq\operatorname{diag}(a_{1}I_{n_{1}},a_{2}I_{n_{2}},\dots,a_{p+1}I_{n_{p+1}}).$

Since
$Q\Lambda_{a}Q^{\scriptscriptstyle\mathsf{T}}=V\Lambda_{a}V^{\scriptscriptstyle\mathsf{T}}$
for $Q,V\in\operatorname{O}_{n}(\mathbb{R})$ if and only if $V=QP$ for some
$P\in\operatorname{S}(\operatorname{O}_{n_{1}}(\mathbb{R})\times\cdots\times\operatorname{O}_{n_{p+1}}(\mathbb{R}))$
if and only if
$(\mathbb{Q}_{1},\dots,\mathbb{Q}_{p})=(\mathbb{V}_{1},\dots,\mathbb{V}_{p})$,
which shows that the map $\varphi_{4}$ is well-defined (and injective too). To
see that $\varphi_{4}$ is a diffeomorphism, observe that $\varphi_{4}$ factors
as

(18)
${{\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)}}$${{\operatorname{Flag}(k_{1},\dots,k_{p};\mathbb{R}^{n})}}$${{\operatorname{SO}_{n}(\mathbb{R})/\operatorname{S}(\operatorname{O}_{n_{1}}(\mathbb{R})\times\cdots\times\operatorname{O}_{n_{p+1}}(\mathbb{R}))}}$$\scriptstyle{\varphi_{4}}$$\scriptstyle{\varphi_{5}}$$\scriptstyle{\varphi_{1}^{-1}\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\scriptscriptstyle\circ$}}}}\varphi_{3}}$

where $\varphi_{5}$ is defined by
$\varphi_{5}(Q\Lambda_{a}Q^{\scriptscriptstyle\mathsf{T}})\coloneqq\llbracket
Q\rrbracket$, $\varphi_{1}$ and $\varphi_{3}$ are the diffeomorphisms defined
earlier in Lemma 3.1. ∎

In the following, let $C$ be as in (15), the subset of $\mathbb{R}^{p+1}$
comprising elements whose coordinates are all distinct. Note that this set
parameterizes all isospectral models of
$\operatorname{Flag}(k_{1},\dots,k_{p};\mathbb{R}^{n})$. The formula for the
transformation between two isospectral models of the same flag manifold or two
quadratic models of the same Grassmannian is straightforward.

######  Proposition 5.2 (Change-of-coordinates for isospectral models).

Let $k_{0},\dots,k_{p+1},n,n_{1},\dots,n_{p+1}$ be as in Theorem 4.1. We use
the parametric characterization in Corollary 4.2 and write $\Lambda_{a}$ as in
(17) for any $(a_{1},\dots,a_{p+1})\in C$. The map

$\varphi:\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)\to\operatorname{Flag}_{b_{1},\dots,b_{p+1}}(k_{1},\dots,k_{p},n),\quad\varphi(Q\Lambda_{a}Q^{\scriptscriptstyle\mathsf{T}})\coloneqq
Q\Lambda_{b}Q^{\scriptscriptstyle\mathsf{T}}$

is an $\operatorname{SO}_{n}(\mathbb{R})$-equivariant diffeomorphism for any
$(a_{1},\dots,a_{p+1})$ and $(b_{1},\dots,b_{p+1})\in C$. In particular, for
positive integers $k\leq n$,

$\varphi:\operatorname{Gr}_{a,b}(k,n)\to\operatorname{Gr}_{c,d}(k,n),\quad\varphi\biggl{(}Q\begin{bmatrix}aI_{k}&0\\\
0&bI_{n-k}\end{bmatrix}Q^{\scriptscriptstyle\mathsf{T}}\biggr{)}\coloneqq
Q\begin{bmatrix}cI_{k}&0\\\
0&dI_{n-k}\end{bmatrix}Q^{\scriptscriptstyle\mathsf{T}},$

is an $\operatorname{SO}_{n}(\mathbb{R})$-equivariant diffeomorphism for any
pairs of distinct real numbers $(a,b)$ and $(c,d)$.

###### Proof.

The same argument used in the proof of Corollary 4.2 shows that $\varphi$ is
well-defined, $\operatorname{SO}_{n}(\mathbb{R})$-equivariant, bijective, and
factors as

${{\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)}}$${{\operatorname{Flag}_{b_{1},\dots,b_{p+1}}(k_{1},\dots,k_{p},n)}}$${{\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n})}}$$\scriptstyle{\varphi}$$\scriptstyle{\varphi_{a}}$$\scriptstyle{\varphi_{b}^{-1}}$

where $\varphi_{a}$ amd $\varphi_{b}$ are the diffeomorphism $\varphi_{4}$ in
Corollary 4.2 with respect to $(a_{1},\dots,a_{p+1})$ and
$(b_{1},\dots,b_{p+1})\in C$. Hence $\varphi$ is a diffeomorphism. ∎

The same proof of Proposition 5.2 may be used to establish a stronger result.

######  Proposition 5.3 (Homotopy for isospectral models).

With notations as in Proposition 5.2, if

(19)
$\operatorname{sign}(a_{i}-a_{j})=\operatorname{sign}(b_{i}-b_{j})\quad\text{for
all }i<j,$

then map

$\displaystyle\Phi_{t}:\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)$
$\displaystyle\to\operatorname{Flag}_{a_{1}+t(b_{1}-a_{1}),\dots,a_{p+1}+t(b_{p+1}-a_{p+1})}(k_{1},\dots,k_{p},n),$
$\displaystyle Q\Lambda_{a}Q^{\scriptscriptstyle\mathsf{T}}$
$\displaystyle\mapsto\Phi_{t}(Q\Lambda_{a}Q^{\scriptscriptstyle\mathsf{T}})\coloneqq\Phi(Q\Lambda_{a}Q^{\scriptscriptstyle\mathsf{T}},t),$

is an $\operatorname{SO}_{n}(\mathbb{R})$-equivariant diffeomorphism for any
$t\in[0,1]$.

###### Proof.

The map $\Phi_{t}$ in Proposition 5.3 is a homotopy along the line segment

$\gamma:[0,1]\to\mathbb{R}^{p+1},\quad\gamma(t)=(1-t)(a_{1},\dots,a_{p+1})+t(b_{1},\dots,b_{p+1}).$

Note that (19) holds if and only if

$(1-t)(a_{i}-a_{j})+t(b_{i}-b_{j})\neq 0\quad\text{for all }t\in[0,1].$

As we saw in the proof of Proposition 4.5, the parameter space
$C\subseteq\mathbb{R}^{p+1}$ is disconnected since it is the complement of
finitely many hyperplanes. For arbitrary $(a_{1},\dots,a_{p+1})$ and
$(b_{1},\dots,b_{p+1})\in C$, it is possible that $\gamma(t)\not\in C$ for
some $t$. The entire curve $\gamma([0,1])\subseteq C$ if and only if
$(a_{1},\dots,a_{p+1})$ and $(b_{1},\dots,b_{p+1})$ are in the same connected
component of $C$, which is in turn equivalent to (19). ∎

##  6\. Equivariant matrix models for Stiefel manifolds

The corresponding results for the Stiefel manifold [24] are considerably
easier. Recall that we write $\operatorname{V}(k,\mathbb{R}^{n})$ for the
abstract Stiefel manifold of orthonormal $k$-frames in $\mathbb{R}^{n}$ and
$\operatorname{V}(k,n)\coloneqq\\{X\in\mathbb{R}^{n\times
k}:X^{\scriptscriptstyle\mathsf{T}}X=I\\}$ for its usual model as $n\times k$
orthonormal matrices.

We will call the following family of minimal
$\operatorname{SO}_{n}(\mathbb{R})$-equivariant models of
$\operatorname{V}(k,\mathbb{R}^{n})$,

(20) $\operatorname{V}_{\\!A}(k,n)\coloneqq\\{Y\in\mathbb{R}^{n\times
k}:Y^{\scriptscriptstyle\mathsf{T}}Y=A\\},$

the _Cholesky models_. Clearly it includes the usual model as
$\operatorname{V}_{\\!I}(k,n)=\operatorname{V}(k,n)$ and is a $G$-manifold via
the action

(21)
$\operatorname{SO}_{n}(\mathbb{R})\times\operatorname{V}_{\\!A}(k,n)\to\operatorname{V}_{\\!A}(k,n),\quad(Q,Y)\mapsto
QY.$

The choice of nomenclature and that it is indeed a model of the Stiefel
manifold will be self-evident after the next two propositions. We begin with
the Stiefel manifold analogue of Theorem 4.1:

######  Proposition 6.1 (Cholesky model I).

Let $k,n\in\mathbb{N}$ with $k\leq n$. If
$\varepsilon:\operatorname{V}(k,\mathbb{R}^{n})\to\mathbb{R}^{n\times k}$ is
an $\operatorname{SO}_{n}(\mathbb{R})$-equivariant embedding, then there is
some $A\in\mathsf{S}^{2}_{\scriptscriptstyle++}(\mathbb{R}^{k})$ such that

$\varepsilon(\operatorname{V}(k,\mathbb{R}^{n}))=\operatorname{V}_{\\!A}(k,n).$

If $n\geq 17$ and $k<(n-1)/2$, then $V_{\\!A}(k,n)$ has the lowest possible
dimension ambient space among all possible
$\operatorname{SO}_{n}(\mathbb{R})$-equivariant models of the Stiefel
manifold.

###### Proof.

Since $\varepsilon$ is $\operatorname{SO}_{n}(\mathbb{R})$-equivariant and
$\operatorname{SO}_{n}(\mathbb{R})$ acts on
$\operatorname{V}(k,\mathbb{R}^{n})$ transitively via (6),
$\varepsilon(\operatorname{V}(k,\mathbb{R}^{n}))$ is an
$\operatorname{SO}_{n}(\mathbb{R})$-orbit. Fix an arbitrary
$Y_{0}\in\varepsilon(\operatorname{V}(k,\mathbb{R}^{n}))$. Then

$\varepsilon(\operatorname{V}(k,\mathbb{R}^{n}))=\left\\{QY_{0}:Q\in\operatorname{SO}_{n}(\mathbb{R})\right\\}\subseteq\\{Y\in\mathbb{R}^{n\times
k}:Y^{\scriptscriptstyle\mathsf{T}}Y=A\\}$

where $A\coloneqq
Y_{0}^{\scriptscriptstyle\mathsf{T}}Y_{0}\in\mathsf{S}^{2}_{\scriptscriptstyle++}(\mathbb{R}^{k})$.
Conversely, if
$Y^{\scriptscriptstyle\mathsf{T}}Y=A=Y_{0}^{\scriptscriptstyle\mathsf{T}}Y_{0}$,
consider the QR decompositions

$Y=Q\begin{bmatrix}R\\\ 0\end{bmatrix},\quad
Y_{0}=Q_{0}\begin{bmatrix}R_{0}\\\ 0\end{bmatrix}$

where $Q,Q_{0}\in\operatorname{O}_{n}(\mathbb{R})$ and
$R,R_{0}\in\operatorname{GL}_{k}(\mathbb{R})$ are upper triangular matrices
with positive diagonals. Thus
$R^{\scriptscriptstyle\mathsf{T}}R=R_{0}^{\scriptscriptstyle\mathsf{T}}R_{0}=A$
are Cholesky decompositions of
$A\in\mathsf{S}^{2}_{\scriptscriptstyle++}(\mathbb{R}^{n})$. By the uniqueness
of the Cholesky decomposition of a symmetric positive definite matrix, we have
$R=R_{0}$ and

$Y=Q\begin{bmatrix}R\\\ 0\end{bmatrix}=Q\begin{bmatrix}R_{0}\\\
0\end{bmatrix}=(QQ_{0}^{\scriptscriptstyle\mathsf{T}})\biggl{(}Q_{0}\begin{bmatrix}R_{0}\\\
0\end{bmatrix}\biggr{)}=(QQ_{0}^{\scriptscriptstyle\mathsf{T}})Y_{0}\in\varepsilon(\operatorname{V}(k,\mathbb{R}^{n})).$

Hence
$\varepsilon(\operatorname{V}(k,\mathbb{R}^{n}))=\operatorname{V}_{\\!A}(k,n)$.
The minimality of $\operatorname{V}_{\\!A}(k,n)$ for $n\geq 17$ and
$k<(n-1)/2$ follows from [17, Proposition 3.8]. ∎

The proof of Proposition 6.1 also yields an alternative parametric
characterization of the Cholesky model, and may be viewed as the Stiefel
manifold analogue of Corollary 4.2.

######  Corollary 6.2 (Cholesky model II).

Let $k,n\in\mathbb{N}$ with $k\leq n$ and
$A\in\mathsf{S}^{2}_{\scriptscriptstyle++}(\mathbb{R}^{n})$. Then

$\operatorname{V}_{\\!A}(k,n)=\biggl{\\{}Q\begin{bmatrix}R\\\
0\end{bmatrix}:Q\in\operatorname{O}_{n}(\mathbb{R})\biggr{\\}}$

where $R\in\operatorname{GL}_{k}(\mathbb{R})$ is the unique Cholesky factor of
$A\in\mathsf{S}^{2}_{\scriptscriptstyle++}(\mathbb{R}^{n})$.

Lastly we present the change-of-coordinate formula for Cholesky models, which
will also provide a pretext for discussing some interesting features of its
parameter space. We borrow the shorthand in [2, Equation 21] and write, for
any $A,B\in\mathsf{S}^{2}_{\scriptscriptstyle++}(\mathbb{R}^{k})$ and
$t\in[0,1]$,

$A\\#_{t}B=A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2}.$

The special case when $t=1/2$ gives
$A\\#_{1/2}B=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}\eqqcolon A\\#B$, the
matrix geometric mean [2, Equation 1]. The Stiefel manifold analogue of
Propositions 5.2 and 5.3 is as follows.

######  Proposition 6.3 (Homotopy and change-of-coordinates for Cholesky
models).

Let $k,n\in\mathbb{N}$ with $k\leq n$. For any
$A,B\in\mathsf{S}^{2}_{\scriptscriptstyle++}(\mathbb{R}^{k})$ and $t\in[0,1]$,

$\displaystyle\psi_{t}:\operatorname{V}_{\\!A}(k,n)$
$\displaystyle\to\operatorname{V}_{\\!A\\#_{t}B}(k,n),$ $\displaystyle Y$
$\displaystyle\mapsto YA^{-1/2}(A^{-1/2}BA^{-1/2})^{t/2}A^{1/2},$

is an $\operatorname{SO}_{n}(\mathbb{R})$-equivariant diffeomorphism. In
particular the map $\psi_{1}$ gives a change-of-coordinates formula from
$\operatorname{V}_{\\!A}(k,n)$ to $\operatorname{V}_{\\!B}(k,n)$.

###### Proof.

The map $\psi_{t}$ is well-defined as

$(YA^{-1/2}(A^{-1/2}BA^{-1/2})^{t/2}A^{1/2})^{\scriptscriptstyle\mathsf{T}}YA^{-1/2}(A^{-1/2}BA^{-1/2})^{t/2}A^{1/2}=A\\#_{t}B$

whenever $Y\in\operatorname{V}_{\\!A}(k,n)$. Since both $A$ and $B$ are
positive definite, $A^{-1/2}(A^{-1/2}BA^{-1/2})^{t/2}A^{1/2}$ is invertible,
so $\psi_{t}$ is a diffeomorphism. It is evidently
$\operatorname{SO}_{n}(\mathbb{R})$-equivariant under (21). ∎

Proposition 6.3 is considerably simpler than Proposition 5.3 as the parameter
space $\mathsf{S}^{2}_{\scriptscriptstyle++}(\mathbb{R}^{n})$ is path
connected and any two points
$A,B\in\mathsf{S}^{2}_{\scriptscriptstyle++}(\mathbb{R}^{n})$ can be connected
by a curve

(22)
$\gamma:[0,1]\to\mathsf{S}^{2}_{\scriptscriptstyle++}(\mathbb{R}^{n}),\quad\gamma(t)=A\\#_{t}B.$

Readers may recognize $\gamma$ as the geodesic curve from $A$ to $B$ in the
Cartan manifold [6, pp. 364–372], i.e.,
$\mathsf{S}^{2}_{\scriptscriptstyle++}(\mathbb{R}^{n})$ equipped with the
Riemannian metric $\mathsf{g}_{A}(X,Y)=\operatorname{tr}(A^{-1}XA^{-1}Y)$, or,
as an abstract manifold, the set of ellipsoids in $\mathbb{R}^{n}$ centered at
the origin. See [20] and [22, Section 3] for a modern exposition, and [15,
Section 8] for further bibliographical references about the Cartan manifold.

##  7\. Equivariant Riemannian metrics

Here we will discuss $\operatorname{SO}_{n}(\mathbb{R})$-invariant Riemannian
metrics for the flag, Grassmann, and Stiefel manifolds that go alongside their
models in this article. As we alluded to at the end of Section 1, each of
these manifolds comes equipped with a god-given Riemannian metric. This is a
result of their structure as $G/H$ with $G$ a compact simple Lie group and $H$
a closed subgroup. Such a $G$ admits a bi-invariant metric, unique up to a
constant factor, which in turn induces an invariant metric on $G/H$. If their
corresponding Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ are related by
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$ where $\mathfrak{m}$ is an
$\operatorname{Ad}_{H}$-invariant subspace of $\mathfrak{g}$, then there is a
one-to-one correspondence

(23) $\\{\text{$G$-invariant metrics on
$G/H$}\\}\longleftrightarrow\\{\text{$H$-invariant metrics on
$\mathfrak{m}$}\\},$

a standard result in differential geometry [7, Proposition 3.16]. In
particular, we have

(24) $\mathfrak{m}\simeq\mathbb{T}_{\llbracket e\rrbracket}G/H,$

and $e\in G$ the identity element.

Slightly less standard (we are unable to find a reference for the simple form
below) is the following simple criterion for an equivariant embedding to be
isometric.

######  Lemma 7.1.

Let $\mathbb{V}$ be a $G$-module and $\varphi:G/H\to\mathbb{V}$ be a
$G$-equivariant embedding. Let $\mathsf{g}$ and $\mathsf{g}^{\prime}$ be
Riemannian metrics on $G/H$ and
$\mathcal{M}\coloneqq\varphi(G/H)\subseteq\mathbb{V}$ respectively. Consider
the linear map

(25) $f:\mathfrak{m}\simeq\mathbb{T}_{\llbracket
e\rrbracket}G/H\xrightarrow{d_{\llbracket
e\rrbracket}\varphi}\mathbb{T}_{\varphi(\llbracket e\rrbracket)}\mathcal{M}.$

Then $\varphi$ is isometric if and only if $f$ is an isometry.

###### Proof.

Since $\varphi$ is $G$-equivariant and $G$ acts on both $G/H$ and
$\mathcal{M}$ transitively, $\varphi$ is isometric if and only if $\varphi$ is
isometric at $\llbracket e\rrbracket$, i.e., the differential map
$d_{\llbracket e\rrbracket}\varphi$ of $\varphi$ at $\llbracket e\rrbracket$
is an isometry. By (23), this is equivalent to $f$ being an isometry. ∎

For the cases of interest to us,

$\displaystyle\operatorname{V}(k,\mathbb{R}^{n})$
$\displaystyle\cong\operatorname{SO}_{n}(\mathbb{R})/\operatorname{SO}_{n-k}(\mathbb{R}),$
$\displaystyle\operatorname{Gr}(k,\mathbb{R}^{n})$
$\displaystyle\cong\operatorname{SO}_{n}(\mathbb{R})/\operatorname{S}(\operatorname{O}_{k}(\mathbb{R})\times\operatorname{O}_{n-k}(\mathbb{R})),$
$\displaystyle\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n})$
$\displaystyle\cong\operatorname{SO}_{n}(\mathbb{R})/\operatorname{S}(\operatorname{O}_{n_{1}}(\mathbb{R})\times\cdots\times\operatorname{O}_{n_{p+1}}(\mathbb{R})).$

We will show that the restriction of the Euclidean inner product on
$\mathbb{R}^{n\times k}$ and $\operatorname{\mathsf{S}}^{2}(\mathbb{R}^{n})$
onto the Cholesky, quadratic, and isospectral models give the Riemannian
metric induced by the bi-invariant metric on
$G=\operatorname{SO}_{n}(\mathbb{R})$ up to a choice of weights.

###  7.1. Quadratic model of the Grassmannian

The bi-invariant metric on $\operatorname{SO}_{n}(\mathbb{R})$ induces an
$\operatorname{SO}_{n}(\mathbb{R})$-invariant metric $\mathsf{g}$ on
$\operatorname{SO}_{n}(\mathbb{R})/\operatorname{S}(\operatorname{O}_{k}(\mathbb{R})\times\operatorname{O}_{n-k}(\mathbb{R}))$,
and thus on $\operatorname{Gr}_{a,b}(k,n)$ via $\varphi_{5}$ in (18), whose
inverse is

(26)
$\varphi_{5}^{-1}:\operatorname{SO}_{n}(\mathbb{R})/\operatorname{S}(\operatorname{O}_{k}(\mathbb{R})\times\operatorname{O}_{n-k}(\mathbb{R}))\to\operatorname{Gr}_{a,b}(k,n),\quad\varphi_{5}^{-1}(\llbracket
Q\rrbracket)=Q\begin{bmatrix}aI_{k}&0\\\
0&bI_{n-k}\end{bmatrix}Q^{\scriptscriptstyle\mathsf{T}}.$

We recall from [27, Proposition 5] that

$\displaystyle\operatorname{\mathsf{\Lambda}}^{2}(\mathbb{R}^{n})$
$\displaystyle=\operatorname{\mathsf{\Lambda}}^{2}(\mathbb{R}^{k})\oplus\operatorname{\mathsf{\Lambda}}^{2}(\mathbb{R}^{n-k})\oplus\mathfrak{m},$
$\displaystyle\mathfrak{m}$
$\displaystyle=\biggl{\\{}B=\begin{bmatrix}0&B_{0}\\\
-B_{0}^{\scriptscriptstyle\mathsf{T}}&0\end{bmatrix}\in\operatorname{\mathsf{\Lambda}}^{2}(\mathbb{R}^{n}):B_{0}\in\mathbb{R}^{k\times(n-k)}\biggr{\\}},$

and that $\mathfrak{m}$ is invariant under conjugation by
$\operatorname{S}(\operatorname{O}_{k}(\mathbb{R})\times\operatorname{O}_{n-k}(\mathbb{R}))$.
The
$\operatorname{S}(\operatorname{O}_{k}(\mathbb{R})\times\operatorname{O}_{n-k}(\mathbb{R}))$-invariant
inner product on $\mathfrak{m}$ is given by

(27) $\langle
B,C\rangle\coloneqq(a-b)^{2}\operatorname{tr}(B^{\scriptscriptstyle\mathsf{T}}C)=2(a-b)^{2}\operatorname{tr}(B_{0}^{\scriptscriptstyle\mathsf{T}}C_{0})$

for any $B,C\in\mathfrak{m}$. This corresponds, via (23), to an
$\operatorname{SO}_{n}(\mathbb{R})$-invariant metric on
$\operatorname{SO}_{n}(\mathbb{R})/\operatorname{S}(\operatorname{O}_{k}(\mathbb{R})\times\operatorname{O}_{n-k}(\mathbb{R}))$
and differs from the god-given metric $\mathsf{g}$ on
$\operatorname{Gr}(k,\mathbb{R}^{n})$ by a weight constant.

The standard Euclidean metric $\operatorname{tr}(XY)$ on
$\mathsf{S}^{2}(\mathbb{R}^{n})$ restricts to a metric $\mathsf{g}^{\prime}$
on $\operatorname{Gr}_{a,b}(k,n)$. We will see that $\mathsf{g}$ and
$\mathsf{g}^{\prime}$ are one and the same, up to a weight constant.

######  Proposition 7.2.

Let
$\operatorname{SO}_{n}(\mathbb{R})/\operatorname{S}(\operatorname{O}_{k}(\mathbb{R})\times\operatorname{O}_{n-k}(\mathbb{R}))$
and $\operatorname{Gr}_{a,b}(k,n)$ be equipped with Riemannian metrics
$\mathsf{g}$ and $\mathsf{g}^{\prime}$ respectively. Then $\varphi_{5}^{-1}$
is an isometric $\operatorname{SO}_{n}(\mathbb{R})$-equivariant
diffeomorphism.

###### Proof.

The $\operatorname{SO}_{n}(\mathbb{R})$-equivariance of $\varphi_{5}^{-1}$ is
evident from its definition. Let $f$ be the linear map in (25) for
$\varphi_{5}^{-1}$. Then by (26),

$f(B)=B\begin{bmatrix}aI_{k}&0\\\
0&bI_{n-k}\end{bmatrix}+\begin{bmatrix}aI_{k}&0\\\
0&bI_{n-k}\end{bmatrix}B^{\scriptscriptstyle\mathsf{T}}=(b-a)B$

for any $B\in\mathfrak{m}$. By (27),

$\langle
B,C\rangle=2(a-b)^{2}\operatorname{tr}(B_{0}^{\scriptscriptstyle\mathsf{T}}C_{0})=\mathsf{g}^{\prime}(f(B),f(C)).$

So $f$ is an isometry and hence so is $\varphi_{5}^{-1}$ by Lemma 7.1. ∎

###  7.2. Cholesky model of the Stiefel manifold

The bi-invariant metric on $\operatorname{SO}_{n}(\mathbb{R})$ induces an
$\operatorname{SO}_{n}(\mathbb{R})$-invariant metric $\mathsf{g}$ on
$\operatorname{SO}_{n}(\mathbb{R})/\operatorname{SO}_{n-k}(\mathbb{R}))$, and
thus on $\operatorname{V}_{\\!A}(k,n)$ via

(28)
$\psi_{A}:\operatorname{SO}_{n}(\mathbb{R})/\operatorname{SO}_{n-k}(\mathbb{R})\to\operatorname{V}_{\\!A}(k,n),\quad\psi_{A}(\llbracket
Q\rrbracket)=Q\begin{bmatrix}R\\\ 0\end{bmatrix},$

which is a diffeomorphism by Proposition 6.1. Here
$R\in\operatorname{GL}_{k}(\mathbb{R})$ is the Cholesky factor of
$A=R^{\scriptscriptstyle\mathsf{T}}R\in\mathsf{S}^{2}_{\scriptscriptstyle++}(\mathbb{R}^{n})$.
We have

$\displaystyle\operatorname{\mathsf{\Lambda}}^{2}(\mathbb{R}^{n})$
$\displaystyle=\operatorname{\mathsf{\Lambda}}^{2}(\mathbb{R}^{n-k})\oplus\mathfrak{m},$
$\displaystyle\mathfrak{m}$
$\displaystyle=\biggl{\\{}\begin{bmatrix}B_{1}&-B_{2}^{\scriptscriptstyle\mathsf{T}}\\\
B_{2}&0\end{bmatrix}\in\mathbb{R}^{n\times
n}:B_{1}\in\operatorname{\mathsf{\Lambda}}^{2}(\mathbb{R}^{k}),\;B_{2}\in\mathbb{R}^{(n-k)\times
k}\biggr{\\}},$

where we identify $B\in\operatorname{\mathsf{\Lambda}}^{2}(\mathbb{R}^{n-k})$
with
$\operatorname{diag}(0,B)\in\operatorname{\mathsf{\Lambda}}^{2}(\mathbb{R}^{n})$.
It is clear that $\mathfrak{m}$ is invariant under conjugation by
$\operatorname{SO}_{n-k}(\mathbb{R})$, where we identify
$Q\in\operatorname{SO}_{n-k}(\mathbb{R})$ with
$\operatorname{diag}(I_{k},Q)\in\operatorname{SO}_{n}(\mathbb{R})$. For
$A=R^{\scriptscriptstyle\mathsf{T}}R\in\mathsf{S}^{2}_{\scriptscriptstyle++}(\mathbb{R}^{n})$,
the $\operatorname{SO}_{n-k}(\mathbb{R})$-invariant $A$-inner product on
$\mathfrak{m}$ is given by

(29) $\langle
B,C\rangle\coloneqq\operatorname{tr}(R^{\scriptscriptstyle\mathsf{T}}(B_{1}^{\scriptscriptstyle\mathsf{T}}C_{1}+B_{2}^{\scriptscriptstyle\mathsf{T}}C_{2})R)$

for any $B,C\in\mathfrak{m}$. This correspond, via (23), to an
$\operatorname{SO}_{n}(\mathbb{R})$-invariant metric $\mathsf{g}_{A}$ on
$\operatorname{SO}_{n}(\mathbb{R})/\operatorname{SO}_{n-k}(\mathbb{R})$ and
differs from the god-given metric $\mathsf{g}$ on
$\operatorname{V}(k,\mathbb{R}^{n})$ by a weight matrix $A$. In particular,
for $A=I$, we have $\mathsf{g}_{I}=\mathsf{g}$.

The standard Euclidean metric
$\operatorname{tr}(X^{\scriptscriptstyle\mathsf{T}}Y)$ on $\mathbb{R}^{n\times
k}$ restricts to a metric $\mathsf{g}^{\prime}$ on
$\operatorname{V}_{\\!A}(k,n)$. We will see that $\mathsf{g}$ and
$\mathsf{g}^{\prime}$ are one and the same, up to a weight matrix.

######  Proposition 7.3 (Riemannian metric on the Stiefel manifold).

Let $\operatorname{SO}_{n}(\mathbb{R})/\operatorname{SO}_{n-k}(\mathbb{R})$
and $\operatorname{V}_{\\!A}(k,n)$ be equipped with Riemannian metrics
$\mathsf{g}_{A}$ and $\mathsf{g}^{\prime}$ respectively. Then $\psi_{A}$ is an
isometric $\operatorname{SO}_{n}(\mathbb{R})$-equivariant diffeomorphism.

###### Proof.

The $\operatorname{SO}_{n}(\mathbb{R})$-equivariance of $\psi_{A}$ is evident
from its definition. Let $f$ be the linear map in (25) for $\psi_{A}$. Then by
(28),

$f(B)=B\begin{bmatrix}R\\\ 0\end{bmatrix}=\begin{bmatrix}B_{1}\\\
-B_{2}^{\scriptscriptstyle\mathsf{T}}\end{bmatrix}R,$

for any $B=\begin{bmatrix}B_{1}&-B_{2}^{\scriptscriptstyle\mathsf{T}}\\\
B_{2}&0\end{bmatrix}\in\mathfrak{m}$. By (29),

$\langle
B,C\rangle=\operatorname{tr}(R^{\scriptscriptstyle\mathsf{T}}(B_{1}^{\scriptscriptstyle\mathsf{T}}C_{1}+B_{2}^{\scriptscriptstyle\mathsf{T}}C_{2})R)=\mathsf{g}^{\prime}(f(B),f(C)).$

So $f$ is an isometry and hence so is $\psi_{A}$ by Lemma 7.1. ∎

###  7.3. Isospectral model of the flag manifold

The argument here is similar to that of Section 7.1, but involves heavier
notations, and as such we think it is instructive to include the special case
in Section 7.1 for clarity. The bi-invariant metric on
$\operatorname{SO}_{n}(\mathbb{R})$ induces an
$\operatorname{SO}_{n}(\mathbb{R})$-invariant metric $\mathsf{g}$ on
$\operatorname{SO}_{n}(\mathbb{R})/\operatorname{S}(\operatorname{O}_{n_{1}}(\mathbb{R})\times\cdots\times\operatorname{O}_{n_{p+1}}(\mathbb{R}))$,
and thus on $\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)$
via $\varphi_{5}$ in (18), whose inverse is

$\varphi_{5}^{-1}:\operatorname{SO}_{n}(\mathbb{R})/\operatorname{S}(\operatorname{O}_{n_{1}}(\mathbb{R})\times\cdots\times\operatorname{O}_{n_{p+1}}(\mathbb{R}))\to\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n),\quad\varphi_{5}^{-1}(\llbracket
Q\rrbracket)\coloneqq Q\Lambda_{a}Q^{\scriptscriptstyle\mathsf{T}},$

where $\Lambda_{a}$ as in (17). We recall from [27, Proposition 5] that

$\displaystyle\operatorname{\mathsf{\Lambda}}^{2}(\mathbb{R}^{n})$
$\displaystyle=\operatorname{\mathsf{\Lambda}}^{2}(\mathbb{R}^{n_{1}})\oplus\dots\oplus\operatorname{\mathsf{\Lambda}}^{2}(\mathbb{R}^{n_{p+1}})\oplus\mathfrak{m},$
$\displaystyle\mathfrak{m}$ $\displaystyle=\\{(B_{ij})\in\mathbb{R}^{n\times
n}:B_{ij}=-B_{ji}^{\scriptscriptstyle\mathsf{T}}\in\mathbb{R}^{n_{i}\times
n_{j}},\;B_{ii}=0,\;1\leq i<j\leq p+1\\},$

and that $\mathfrak{m}$ is invariant under conjugation by
$\operatorname{S}(\operatorname{O}_{n_{1}}(\mathbb{R})\times\cdots\times\operatorname{O}_{n_{p+1}}(\mathbb{R}))$.
The
$\operatorname{S}(\operatorname{O}_{n_{1}}(\mathbb{R})\times\cdots\times\operatorname{O}_{n_{p+1}}(\mathbb{R}))$-invariant
inner product on $\mathfrak{m}$ is given by

(30) $\langle B,C\rangle\coloneqq 2\sum_{1\leq i<j\leq
p+1}(a_{i}-a_{j})^{2}\operatorname{tr}(B_{ij}^{\scriptscriptstyle\mathsf{T}}C_{ij}),$

where
$B,C\in\mathfrak{m}\subseteq\operatorname{\mathsf{\Lambda}}^{2}(\mathbb{R}^{n})$
are partitioned as $B=(B_{ij})$, $C=(C_{ij})$ with $B_{ij}$,
$C_{ij}\in\mathbb{R}^{n_{i}\times n_{j}}$ for $i,j\in\\{1,\dots,p+1\\}$. This
corresponds via (23) to an $\operatorname{SO}_{n}(\mathbb{R})$-invariant
metric $\mathsf{g}_{a}$ on
$\operatorname{SO}_{n}(\mathbb{R})/\operatorname{S}(\operatorname{O}_{n_{1}}(\mathbb{R})\times\cdots\times\operatorname{O}_{n_{p+1}}(\mathbb{R}))$
and differs from the god-given metric $\mathsf{g}$ on
$\operatorname{Flag}(k_{1},\dots,k_{p},\mathbb{R}^{n})$ by a weight vector
$a\coloneqq(a_{1},\dots,a_{p+1})$.

The standard Euclidean metric $\operatorname{tr}(XY)$ on
$\mathsf{S}^{2}(\mathbb{R}^{n})$ when restricted to
$\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)$ gives a
metric $\mathsf{g}^{\prime}$. We will see that $\mathsf{g}$ and
$\mathsf{g}^{\prime}$ are one and the same, up to a weight vector.

######  Proposition 7.4 (Riemannian metric on the flag manifold).

Let
$\operatorname{SO}_{n}(\mathbb{R})/\operatorname{S}(\operatorname{O}_{n_{1}}(\mathbb{R})\times\dots\times\operatorname{O}_{n_{p+1}}(\mathbb{R}))$
and $\operatorname{Flag}_{a_{1},\dots,a_{p+1}}(k_{1},\dots,k_{p},n)$ be
equipped be Riemannian metrics $\mathsf{g}_{a}$ and $\mathsf{g}^{\prime}$
respectively. Then $\varphi_{5}^{-1}$ is an isometric
$\operatorname{SO}_{n}(\mathbb{R})$-equivariant diffeomorphism.

###### Proof.

The $\operatorname{SO}_{n}(\mathbb{R})$-equivariance of $\varphi_{5}^{-1}$ is
evident from its definition. Let $f$ be the linear map defined in Lemma 7.1
for $\varphi_{5}^{-1}$. Then by (18),

$f(B)=B\Lambda_{a}+\Lambda_{a}B^{\scriptscriptstyle\mathsf{T}}=[(a_{j}-a_{i})B_{ij}]_{i,j=1}^{p+1}$

for any $B\in\mathfrak{m}$. By (30),

$\langle B,C\rangle=2\sum_{1\leq i<j\leq
p+1}(a_{i}-a_{j})^{2}\operatorname{tr}(B_{ij}^{\scriptscriptstyle\mathsf{T}}C_{ij})=\mathsf{g}^{\prime}(f(B),f(C))$

So $f$ is an isometry and hence so is $\varphi_{5}^{-1}$ by Lemma 7.1. ∎

##  8\. Conclusion

We used to be able to count on one hand the number of different models for
each of these manifolds. With these families of models, we now have
uncountably many choices, and having such flexibility can provide a real
benefit as different models are useful in different ways.

Take the family of quadratic models $\operatorname{Gr}_{a,b}(k,n)$ for
example. The traceless model with $(a,b)=(n-k,k)$ in Corollary 4.3 has the
lowest dimension but the involution model with $(a,b)=(1,-1)$ in [14] has the
best condition number. It may appear that the projection model with
$(a,b)=(1,0)$ makes the worst choice from a computational perspective since it
is, up to a constant, the only model in the family with singular matrices.
However we found in [15] that it is the most suitable model for discussing
computational complexity issues, as many well-known NP-hard problems have
natural formulations as optimization problems in the projection model.

We hope that these families of models for various manifolds described and
classified in this article would provide useful computational platforms for
practical applications involving these manifolds.

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