# Eigenvalues and Singular Value Decomposition of Dual Complex Matrices

Liqun Qi111Department of Applied Mathematics, The Hong Kong Polytechnic
University, Hung Hom, Kowloon, Hong Kong (maqilq@polyu.edu.hk). and Ziyan
Luo222Department of Mathematics, Beijing Jiaotong University, Beijing 100044,
China. (zyluo@bjtu.edu.cn). This author’s work was supported by NSFC (Grant
No. 11771038) and Beijing Natural Science Foundation (Grant No. Z190002).

###### Abstract

We introduce right eigenvalues and subeigenvalues for square dual complex
matrices. An $n\times n$ dual complex Hermitian matrix has exactly $n$ right
eigenvalues and subeigenvalues, which are all real. The Hermitian matrix is
positive semi-definite or definite if and only if all of its right eigenvalues
and subeigenvalues are nonnegative or positive, respectively. A Hermitian
matrix can be diagonalized if and only if it has no right subeigenvalues. Then
we present the singular value decomposition for general dual complex matrices.
The results are further extended to dual quaternion matrices.

Key words. Dual complex matrices, Clifford algebra, right eigenvalues, right
subeigenvalues, singular value decomposition

##  1 Introduction

In 1873, W.K. Clifford [7] introduced dual numbers, dual complex numbers and
dual quaternions. These become the core knowledge of Clifford algebra or
geometric algebra.

Dual numbers, dual complex numbers and dual quaternions now have wide
applications in automatic differentiation, mechanics, geometry, rigid body
motions, robotics and computer graphics [1, 4, 6, 8, 10, 14, 17].

On the other hand, quaternions found wide applications in color image and
video processing [5, 13, 15, 19]. In these applications, quaternion matrices
play a significant role. The knowledge about right eigenvalues of quaternion
has been developed for several decades [2, 15, 18, 21]. In particular, Zhang
[21] presented singular value decomposition for quaternion matrices. This work
is widely cited in applications [5, 13, 15, 19].

Thus, we wonder if there are counter parts for dual number matrices, dual
complex matrices and dual quaternion matrices.

Recently, Gutin [11] presented singular value decomposition for dual number
matrices. There is also study on dual complex matrices [3]. Thus, in this
paper, we study dual complex matrices.

There are two difficulties in dual complex numbers. They are non-communitative
and there are zero divisors. This makes dual complex matrix analysis more
difficult, comparing with dual number matrix analysis and quaternion matrix
analysis.

In the next section, we present some basic knowledge on dual complex numbers
and dual complex matrices.

Then we define right eigenvalues for square dual complex matrices in Section
3. We show that the standard part of a right eigenvalue of a square dual
complex matrix and the standard part of the corresponding right eigenvector
are exactly a complex eigenvalue and its corresponding complex eigenvector of
the standard part of the dual complex matrix. This indicates that an $n\times
n$ dual complex matrix has at most $n$ complex right eigenvalues. We give a
necessary and sufficient condition for a dual complex number to be a right
eigenvalue of a square dual complex matrix.

In Section 4, we first observe that the standard part of a dual complex
Hermitian matrix is a complex Hermitian matrix, and the infinitesimal part is
a complex skew symmetric matrix. Then we show that the right eigenvalues of a
dual complex Hermitian matrix are real. Thus, an $n\times n$ dual complex
Hermitian matrix has at most $n$ right eigenvalues. In particular, a simple
eigenvalue of the standard part of a dual complex matrix is a right eigenvalue
of that dual complex Hermitian matrix. An $n\times n$ dual complex Hermitian
matrix is diagonalizable if and only if it has $n$ right eigenvalues.

Then in Section 5, we define right subeigenvalues for square dual complex
matrices. There are two right subeigenvectors associated with a right
subeigenvalue. These two right subeigenvectors are orthogonal to each other.
There is also a nonzero adjoint parameter associated with them. If that
adjoint parameter is degenerated to zero, then the right subeigenvalue was
reduced to a right eigenvalue of multiplicity $2$. We show that the right
subeigenvalues of a dual complex Hermitian matrix are also real, and are
multiple eigenvalues of the standard part of that dual complex Hermitian
matrix. Then we show that an $n\times n$ dual complex Hermitian matrix has
exactly $n$ right eigenvalues and subeigenvalues.

In Section 6, we present the singular value decomposition for general dual
complex matrices.

Our work lays the foundation of dual complex matrix analysis, and paves the
way for further study on dual quaternion matrices and their applications. In
Section 7, we briefly study extensions of our results to dual quaternion
matrices. Some final remarks are made in Section 8.

##  2 Dual Complex Numbers and Dual Complex Matrices

###  2.1 Dual Complex Numbers

We denote the real numbers, the complex numbers, the dual numbers and the dual
complex numbers by ${\mathbb{R}}$, $\mathbb{C}$, ${\mathbb{D}}$ and
$\mathbb{DC}$, respectively. Scalars, vectors and matrices are denoted by
small letters, bold small letters and capital letters, respectively. A dual
complex number $q$ has the form

$q=q_{0}+q_{1}\mathbf{i}+q_{2}\epsilon\mathbf{j}+q_{3}\epsilon\mathbf{k},$

where $q_{0},q_{1},q_{2}$ and $q_{3}$ are real numbers,
$\mathbf{i},\mathbf{j}$ and $\mathbf{k}$ are three imaginary units of
quaternions, satisfying

$\mathbf{i}^{2}=\mathbf{j}^{2}=\mathbf{k}^{2}=\mathbf{i}\mathbf{j}\mathbf{k}=-1,$
$\mathbf{i}\mathbf{j}=-\mathbf{j}\mathbf{i}=\mathbf{k},\
\mathbf{j}\mathbf{k}=-\mathbf{k}\mathbf{j}=\mathbf{i},\mathbf{k}\mathbf{i}=-\mathbf{i}\mathbf{k}=\mathbf{j},$

and $\epsilon$ is the infinitesimal unit, satisfying $\epsilon^{2}=0$. These
rules, along with the distribution law, determine the product of two dual
complex numbers. Unlike multiplication of dual numbers or of complex numbers,
the multiplication of dual complex numbers is noncommutative. Let
$q=q_{0}+q_{1}\mathbf{i}+q_{2}\epsilon\mathbf{j}+q_{3}\epsilon\mathbf{k}$ as
defined above, and

$p=p_{0}+p_{1}\mathbf{i}+p_{2}\epsilon\mathbf{j}+p_{3}\epsilon\mathbf{k},$

where $p_{0},p_{1},p_{2}$ and $p_{3}$ are real numbers. Then

$pq=(p_{0}q_{0}-p_{1}q_{1})+(p_{0}q_{1}+p_{1}q_{0})\mathbf{i}+(p_{0}q_{2}+p_{2}q_{0}-p_{1}q_{3}+p_{3}q_{1})\epsilon\mathbf{j}+(p_{0}q_{3}+p_{3}q_{0}+p_{1}q_{2}-p_{2}q_{1})\epsilon\mathbf{k},$
$qp=(p_{0}q_{0}-p_{1}q_{1})+(p_{0}q_{1}+p_{1}q_{0})\mathbf{i}+(p_{0}q_{2}+p_{2}q_{0}+p_{1}q_{3}-p_{3}q_{1})\epsilon\mathbf{j}+(p_{0}q_{3}+p_{3}q_{0}-p_{1}q_{2}+p_{2}q_{1})\epsilon\mathbf{k}.$

In general, $pq\not=qp$.

The conjugate of
$q=q_{0}+q_{1}\mathbf{i}+q_{2}\epsilon\mathbf{j}+q_{3}\epsilon\mathbf{k}$ is

$\bar{q}=q_{0}-q_{1}\mathbf{i}-q_{2}\epsilon\mathbf{j}-q_{3}\epsilon\mathbf{k}.$

The magnitude of $q$ is

$|q|=\sqrt{q_{0}^{2}+q_{1}^{2}}.$

We see that

$q\bar{q}=\bar{q}q=|q|^{2},$ $|pq|=|qp|=|p|\cdot|q|,$

and

$\overline{pq}=\bar{q}\bar{p}.$

A dual complex number $q$ is a real number if and only if $q=\bar{q}$. The
standard part of $q$ is $q_{st}=q_{0}+q_{1}\mathbf{i}$. The infinitesimal part
of $q$ is $q_{\mathcal{I}}=q_{2}+q_{3}\mathbf{i}$. Note that both $q_{st}$ and
$q_{\mathcal{I}}$ are complex numbers, and

$q=q_{st}+q_{\mathcal{I}}\epsilon\mathbf{j}.$

We say that a dual complex number $q$ is appreciable if its standard part is
nonzero. Otherwise, we say that it is infinitesimal.

It follows the multiplicative inverse of an appreciable dual complex number
$q$ is given by

$q^{-1}={\bar{q}\over|q|^{2}}.$

Two dual complex numbers $p$ and $q$ are said to be similar if there exists an
appreciable dual complex number $u$ such that $u^{-1}pu=q$. This is denoted as
$p\sim q$. Then $\sim$ is an equivalence relation. Denote the equivalence
class containing $q$ by $[q]$. If $p\sim q$, then $|p|=|q|$. Thus, if $p\sim
q$, then either both $p$ and $q$ are appreciable, or both of them are
infinitesimal.

The primary application of dual complex numbers is in representing rigid body
motions in 2D space. The set of dual complex numbers is a subset of the set of
dual quaternions. For more properties and applications of dual complex
numbers, see [10, 14]. In [14], such dual complex numbers are called anti-
commutative dual complex numbers, which parametrize two dimension rotation and
translation together. With this presentation, we can easily interpolate or
blend two or more rigid transformations at a low computation cost. In this
paper, we simply call them dual complex numbers.

###  2.2 Dual Complex Matrices

The collections of real, complex and dual complex $m\times n$ matrices are
denoted by ${\mathbb{R}}^{m\times n}$, ${\mathbb{C}}^{m\times n}$ and
${\mathbb{DC}}^{m\times n}$, respectively.

A dual complex matrix $A=(a_{ij})\in{\mathbb{DC}}^{m\times n}$ can be denoted
as

$A=A_{0}+A_{1}\mathbf{i}+A_{2}\epsilon\mathbf{j}+A_{3}\epsilon\mathbf{k},$ (1)

where $A_{0},A_{1},A_{2},A_{3}\in{\mathbb{R}}^{m\times n}$. The transpose of
$A$ is $A^{\top}=(a_{ji})$. The conjugate of $A$ is $\bar{A}=(\bar{a}_{ij})$.
The conjugate transpose of $A$ is $A^{*}=(\bar{a}_{ji})=\bar{A}^{\top}$.

For $A\in{\mathbb{DC}}^{m\times n}$, expressed by (1), its standard part is
$A_{st}\equiv{\rm st}(A)=A_{0}+A_{1}\mathbf{i}$, its infinitesimal part is
$A_{\mathcal{I}}\equiv\mathcal{I}(A)=A_{2}+A_{3}\mathbf{i}$. Both $A_{st}$ and
$A_{\mathcal{I}}$ are complex matrices, and

$A=A_{st}+A_{\mathcal{I}}\epsilon\mathbf{j}.$

The Frobenius norm of $A$ is

$\|A\|_{F}=\sqrt{\sum_{i=1}^{m}\sum_{j=1}^{n}|a_{ij}|^{2}}.$

Let $A\in{\mathbb{DC}}^{m\times n}$ and $B\in{\mathbb{DC}}^{n\times r}$. Then
we have $(AB)^{*}=B^{*}A^{*}$. But in general,
$(AB)^{\top}\not=B^{\top}A^{\top}$ and $\overline{AB}\not=\bar{A}\bar{B}$ in
general.

A square dual complex matrix $A\in{\mathbb{DC}}^{n\times n}$ is called
Hermitian if $A^{*}=A$; unitary if $A^{*}A=I$; and invertible (nonsingular) if
$AB=BA=I$ for some $B\in{\mathbb{DC}}^{n\times n}$. We have
$(AB)^{-1}=B^{-1}A^{-1}$ if $A$ and $B$ are invertible, and
$\left(A^{*}\right)^{-1}=\left(A^{-1}\right)^{*}$ if $A$ is invertible.

Applications of dual complex matrices include classical mechanics and
robotics, complex representations of the Lorentz group in relativity and
electrodynamics, conformal mappings in computer vision, the physics of
scattering processes, etc., see [3].

###  2.3 Dual Complex Vectors

Denote $\mathbf{x}\in{\mathbb{DC}}^{n\times 1}$ and
$\mathbf{x}^{\top}\in{\mathbb{DC}}^{1\times n}$ for column and row dual
complex vectors. We say that $\mathbf{x}\in{\mathbb{DC}}^{n\times 1}$ is
appreciable if at least one of its component is appreciable. The magnitude of
$\mathbf{x}\in{\mathbb{DC}}^{n\times 1}$ is

$\|\mathbf{x}\|=\sqrt{\mathbf{x}^{*}\mathbf{x}}.$

If $\|\mathbf{x}\|=1$, then we say that $\mathbf{x}$ is a unit column vector.
If $\mathbf{x},\mathbf{y}\in{\mathbb{DC}}^{n\times 1}$ and
$\mathbf{x}^{*}\mathbf{y}=0$, then we say that $\mathbf{x}$ and $\mathbf{y}$
are orthogonal to each other. If
$\mathbf{x}^{(1)},\cdots,\mathbf{x}^{(n)}\in{\mathbb{DC}}^{n\times 1}$ and
$\left(\mathbf{x}^{(i)}\right)^{*}\mathbf{x}^{(j)}=\delta_{ij}$ for
$i,j=1,\cdots,n$, where $\delta_{ij}$ is the Kronecker symbol, then we say
that $\\{\mathbf{x}^{(1)},\cdots,\mathbf{x}^{(n)}\\}$ is an orthonormal basis
of ${\mathbb{DC}}^{n\times 1}$. A square dual complex matrix
$A\in{\mathbb{DC}}^{n\times n}$ is unitary if and only if its column vectors
form an orthonormal basis of ${\mathbb{DC}}^{n\times 1}$.

The following properties for complex matrices still hold for dual complex
matrices:

1\. If $A\in{\mathbb{DC}}^{n\times n}$ and
$\mathbf{x}\in{\mathbb{DC}}^{n\times 1}$, then

$\|A\mathbf{x}\|\leq\|A\|_{F}\|\mathbf{x}\|.$

2\. If $U\in{\mathbb{DC}}^{n\times n}$ is unitary and
$\mathbf{x}\in{\mathbb{DC}}^{n\times 1}$, then

$\|U\mathbf{x}\|=\|\mathbf{x}\|.$

3\. (Unitary Invariance) If $U\in{\mathbb{DC}}^{m\times m}$ and
$V\in{\mathbb{DC}}^{n\times n}$ are unitary, and $A\in{\mathbb{DC}}^{m\times
n}$, then

$\|UAV\|_{F}=\|A\|_{F}.$

The proof of the unitary invariance property is the same as the proof of
orthogonal invariance in [9].

Note that if both $\mathbf{x}$ and $q$ are appreciable, then $\mathbf{x}q$ is
appreciable.

The following proposition can be proved by definition directly.

######  Proposition 2.1.

Suppose that $A\in{\mathbb{DC}}^{n\times n}$ is invertible. Then all column
and row vectors of $A$ are appreciable.

##  3 Right Eigenvalues of Dual Complex Matrices

Suppose that $A\in{\mathbb{DC}}^{n\times n}$. If there are
$\lambda\in\mathbb{DC}$ and $\mathbf{x}\in{\mathbb{DC}}^{n\times 1}$, where
$\mathbf{x}$ is appreciable, such that

$A\mathbf{x}=\mathbf{x}\lambda,$ (2)

then we say that $\lambda$ is a right eigenvalue of $A$, with $\mathbf{x}$ as
a corresponding right eigenvector.

Here, we request that $\mathbf{x}$ is appreciable. See the definition of
eigenvectors of dual number matrices in [11].

Note that $A\mathbf{x}=\mathbf{x}\lambda$ implies
$A(\mathbf{x}q)=(A\mathbf{x})q=\mathbf{x}\lambda q=(\mathbf{x}q)(q^{-1}\lambda
q)$ if $q$ is appreciable. Thus, if $\lambda$ is a right eigenvalue of $A$,
then any member of $[\lambda]$ is a right eigenvalue of $A$.

We may also define left eigenvalues, but we do not go to this direction, as it
is not related with our discussion.

We prove a lemma.

######  Lemma 3.1.

Let $\mathbf{a}\in\mathbb{C}^{n}$. Then
$\mathbf{j}\mathbf{a}=\bar{\mathbf{a}}\mathbf{j}$.

###### Proof.

Let $\mathbf{a}=\mathbf{b}+\mathbf{c}\mathbf{i}$, where $\mathbf{b}$ and
$\mathbf{c}$ are real vectors. Then

$\mathbf{j}\mathbf{a}=\mathbf{b}\mathbf{j}-\mathbf{c}\mathbf{k}=(\mathbf{b}-\mathbf{c}\mathbf{i})\mathbf{j}=\bar{\mathbf{a}}\mathbf{j}.$

∎

We now establish conditions for a dual complex number to be a right eigenvalue
of a square dual complex matrix.

We have the following theorem.

######  Theorem 3.2.

Suppose that
$A=A_{st}+A_{\mathcal{I}}\epsilon\mathbf{j}\in{\mathbb{DC}}^{n\times n}$. Then
$\lambda=\lambda_{st}+\lambda_{\mathcal{I}}\epsilon\mathbf{j}$ is a right
eigenvalue of $A$ with a right eigenvector
$\mathbf{x}=\mathbf{x}_{st}+\mathbf{x}_{I}\epsilon\mathbf{j}$ only if
$\lambda_{st}$ is an eigenvalue of the complex matrix $A_{st}$ with an
eigenvector $\mathbf{x}_{st}$, i.e., $\mathbf{x}_{st}\not=\mathbf{0}$ and

$A_{st}\mathbf{x}_{st}=\lambda_{st}\mathbf{x}_{st}.$ (3)

Furthermore, if $\lambda_{st}$ is an eigenvalue of the complex matrix $A_{st}$
with an eigenvector $\mathbf{x}_{st}$, then $\lambda$ is a right eigenvalue of
$A$ with a right eigenvector $\mathbf{x}$ if and only if
$\lambda_{\mathcal{I}}$ and $\mathbf{x}_{\mathcal{I}}$ satisfy

$\lambda_{\mathcal{I}}\mathbf{x}_{st}=A_{\mathcal{I}}\bar{\mathbf{x}}_{st}+A_{st}\mathbf{x}_{\mathcal{I}}-\bar{\lambda}_{st}\mathbf{x}_{\mathcal{I}}.$
(4)

###### Proof.

By definition, $\lambda$ is a right eigenvalue of $A$ with a right eigenvector
$\mathbf{x}$ if and only if $\mathbf{x}_{st}\not=\mathbf{0}$ and
$A\mathbf{x}=\mathbf{x}\lambda$. Then $A\mathbf{x}=\mathbf{x}\lambda$ is
equivalent to

$(A_{st}+A_{\mathcal{I}}\epsilon\mathbf{j})(\mathbf{x}_{st}+\mathbf{x}_{I}\epsilon\mathbf{j})=(\mathbf{x}_{st}+\mathbf{x}_{I}\epsilon\mathbf{j})(\lambda_{st}+\lambda_{\mathcal{I}}\epsilon\mathbf{j}).$

This is further equivalent to
$A_{st}\mathbf{x}_{st}=\mathbf{x}_{st}\lambda_{st}$, i.e., (3), and

$A_{st}\mathbf{x}_{\mathcal{I}}\epsilon\mathbf{j}+A_{\mathcal{I}}\epsilon\mathbf{j}\mathbf{x}_{st}=\mathbf{x}_{st}\lambda_{\mathcal{I}}\epsilon\mathbf{j}+\mathbf{x}_{\mathcal{I}}\epsilon\mathbf{j}\lambda_{st}.$
(5)

By Lemma 3.1, (5) is equivalent to

$A_{st}\mathbf{x}_{\mathcal{I}}\mathbf{j}+A_{\mathcal{I}}\bar{\mathbf{x}}_{st}\mathbf{j}=\mathbf{x}_{st}\lambda_{\mathcal{I}}\mathbf{j}+\mathbf{x}_{\mathcal{I}}\bar{\lambda}_{st}\mathbf{j},$

which is further equivalent to (4). The conclusions of this theorem follow
from these. ∎

Let $\lambda\equiv\lambda_{st}$ and $\lambda_{\mathcal{I}}=0$ in this theorem.
We have the following corollary.

######  Corollary 3.3.

Suppose that
$A=A_{st}+A_{\mathcal{I}}\epsilon\mathbf{j}\in{\mathbb{DC}}^{n\times n}$. Then
a complex number $\lambda$ is a right eigenvalue of $A$ with a right
eigenvector $\mathbf{x}_{st}+\mathbf{x}_{I}\epsilon\mathbf{j}$ only if
$\lambda$ is an eigenvalue of the complex matrix $A_{st}$ with an eigenvector
$\mathbf{x}_{st}$, i.e., $\mathbf{x}_{st}\not=\mathbf{0}$ and

$A_{st}\mathbf{x}_{st}=\lambda\mathbf{x}_{st}.$ (6)

This indicates that $A$ has at most $n$ complex right eigenvalues.
Furthermore, if $\lambda$ is an eigenvalue of the complex matrix $A_{st}$ with
an eigenvector $\mathbf{x}_{st}$, then $\lambda$ is a right eigenvalue of $A$
with a right eigenvector $\mathbf{x}$ if and only if
$\mathbf{x}_{\mathcal{I}}$ satisfies

$A_{\mathcal{I}}\bar{\mathbf{x}}_{st}+A_{st}\mathbf{x}_{\mathcal{I}}-\bar{\lambda}\mathbf{x}_{\mathcal{I}}=\mathbf{0}.$
(7)

A square quaternion matrix always has complex right eigenvalues [2, 18, 21]. A
square dual complex matrix may have no complex right eigenvalue at all. This
is very different. See the following example.

Example 1 Let $A=I_{n}+I_{n}\epsilon\mathbf{j}$, i.e.,
$A_{st}=A_{\mathcal{I}}=I_{n}$. Then the only eigenvalue of $A_{st}$ is
$\lambda_{st}=1$ with multiplicity $n$. Then (7) is equivalent to
$\bar{\mathbf{x}}_{st}=\mathbf{0}$, which contradicts that
$\mathbf{x}_{st}\not=\mathbf{0}$. Thus, $A$ has no complex right eigenvalue.
However, let $\mathbf{x}={\bf e}+{\bf e}\mathbf{i}$, where ${\bf
e}=(1,\cdots,1)^{\top}$, and

$\lambda=1-\mathbf{i}\epsilon\mathbf{j}.$

Then we see that $\lambda$ is a right eigenvalue of $A$ with $\mathbf{x}$ as
its right eigenvector.

In the next section, we will give an example that a dual complex Hermitian
matrix has no right eigenvalue at all.

##  4 Right Eigenvalues and Right Eigenvectors of Hermitian Matrices

Suppose that $A\in{\mathbb{DC}}^{n\times n}$ is a Hermitian matrix. Then for
$\mathbf{x}\in{\mathbb{DC}}^{n\times 1}$,
$\mathbf{x}^{*}A\mathbf{x}=(\mathbf{x}^{*}A\mathbf{x})^{*}$ is a real number.
We say that $A$ is positive semi-definite if for all
$\mathbf{x}\in{\mathbb{DC}}^{n\times 1}$, $\mathbf{x}^{*}A\mathbf{x}\geq 0$.
We say that $A$ is positive definite if for all
$\mathbf{x}\in{\mathbb{DC}}^{n\times 1}$ and $\mathbf{x}$ is appreciable,
$\mathbf{x}^{*}A\mathbf{x}>0$.

By definition, we have the following proposition.

######  Proposition 4.1.

A dual complex matrix
$A=A_{st}+A_{\mathcal{I}}\epsilon\mathbf{j}\in{\mathbb{DC}}^{n\times n}$ is a
Hermitian matrix if and only if $A_{st}$ is a complex Hermitian matrix and
$A_{\mathcal{I}}$ is a skew-symmetric complex matrix, i.e.,
$A_{st}^{*}=A_{st}$ and $A_{\mathcal{I}}^{\top}=-A_{\mathcal{I}}$.

Then we have the following proposition.

######  Proposition 4.2.

A right eigenvalue $\lambda$ of a Hermitian matrix
$A=A_{st}+A_{\mathcal{I}}\epsilon\mathbf{j}\in{\mathbb{DC}}^{n\times n}$ must
be a real number and is an eigenvalue of the complex Hermitian matrix
$A_{st}$. Thus, a dual complex Hermitian matrix has at most $n$ real right
eigenvalues and no other right eigenvalues.

A right eigenvalue of a positive semi-definite Hermitian matrix
$A\in{\mathbb{DC}}^{n\times n}$ must be a nonnegative number. Thus, a dual
complex positive semi-definite Hermitian matrix has at most $n$ nonnegative
right eigenvalues and no other right eigenvalues.

A right eigenvalue of a positive definite Hermitian matrix
$A\in{\mathbb{DC}}^{n\times n}$ must be a positive number. Thus, a dual
complex positive definite Hermitian matrix has at most $n$ positive right
eigenvalues and no other right eigenvalues.

###### Proof.

Suppose that $A\in{\mathbb{DC}}^{n\times n}$ is a Hermitian matrix, $\lambda$
is a right eigenvalue of $A$ with right eigenvectors $\mathbf{x}$. Then

$\mathbf{x}^{*}A\mathbf{x}=\mathbf{x}^{*}\mathbf{x}\lambda=\|\mathbf{x}\|^{2}\lambda$

is a real number. Thus, $\lambda$ is a real number. By Corollary 3.3 and
Proposition 4.1, $\lambda$ is an eigenvalue of the complex Hermitian matrix
$A_{st}$. Thus, a dual complex Hermitian matrix has at most $n$ real right
eigenvalues and no other right eigenvalues.

The other conclusions can be proved similarly. ∎

We now give an example that a dual complex Hermitian matrix has no right
eigenvalue at all.

Example 2 Let $n=2$ and

$A=\begin{bmatrix}1&\epsilon\mathbf{j}\\\ -\epsilon\mathbf{j}&1\end{bmatrix}.$

By definition, $A$ is a Hermitian matrix. Denote

$A_{st}=I_{2}=\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}\ {\rm and}\
A_{\mathcal{I}}=\begin{bmatrix}0&1\\\ -1&0\end{bmatrix}.$

Suppose that $A$ has a right eigenvalue $\lambda$. By Proposition 4.2,
$\lambda$ is a real number and is an eigenvalue of $I_{2}$. Thus, $\lambda=1$.
Assume it has a right eigenvector
$\mathbf{x}=\mathbf{x}_{st}+\mathbf{x}_{\mathcal{I}}\epsilon\mathbf{j}$. By
Corollary 3.3, it needs to satisfy (7), i.e.,
$A_{\mathcal{I}}\bar{\mathbf{x}}_{st}=\mathbf{0}$. This implies that
$\mathbf{x}_{st}=\mathbf{0}$. Then $\mathbf{x}$ is not appreciable, and is not
a right eigenvector. Therefore, $A$ has no right eigenvalue at all.

Fortunately, the above example only occurs in the case that $A_{st}$ has
multiple eigenvalues. Recall that an eigenvalue with multiplicity $1$ is
called a simple eigenvalue.

######  Theorem 4.3.

Let $A=A_{st}+A_{\mathcal{I}}\epsilon\mathbf{j}\in{\mathbb{DC}}^{n\times n}$
be a Hermitian matrix. If $\lambda$ is a simple eigenvalue of $A_{st}$, then
$\lambda$ is a right eigenvalue of $A$.

###### Proof.

By Proposition 4.1, $A_{st}$ is a complex Hermitian matrix and $\lambda$ is
real. Then (7) becomes

$A_{\mathcal{I}}\bar{\mathbf{x}}_{st}=(\lambda
I_{n}-A_{st})\mathbf{x}_{\mathcal{I}}.$ (8)

Since $\lambda$ is a simple eigenvalue of $A_{st}$, (8) has a solution
$\mathbf{x}_{\mathcal{I}}$ if and only if

$\mathbf{x}_{st}^{*}A_{\mathcal{I}}\bar{\mathbf{x}}_{st}=0.$

But

$\mathbf{x}_{st}^{*}A_{\mathcal{I}}\bar{\mathbf{x}}_{st}=\left(\mathbf{x}_{st}^{*}A_{\mathcal{I}}\bar{\mathbf{x}}_{st}\right)^{\top}=\mathbf{x}_{st}^{*}(A_{\mathcal{I}})^{\top}\bar{\mathbf{x}}_{st}=-\mathbf{x}_{st}^{*}A_{\mathcal{I}}\bar{\mathbf{x}}_{st},$

as by Proposition 4.1, $A_{\mathcal{I}}^{\top}=-A_{\mathcal{I}}$. Thus,

$\mathbf{x}_{st}^{*}A_{\mathcal{I}}\bar{\mathbf{x}}_{st}=0.$

This implies (8) has a solution $\mathbf{x}_{\mathcal{I}}$. By Corollary 3.3,
$\lambda$ is a right eigenvalue of $A$. ∎

We now consider right eigenvectors of a dual complex Hermitian matrix.

######  Proposition 4.4.

Suppose that $A\in{\mathbb{DC}}^{n\times n}$ is a Hermitian matrix, and has a
right eigenvalue $\lambda$, with right eigenvectors
$\mathbf{x}^{(1)},\cdots,\mathbf{x}^{(k)}$. Then
$\mathbf{y}=\sum_{j=1}^{k}\mathbf{x}^{(j)}\alpha_{j}$ is a right eigenvector
of $A$, associated with $\lambda$, as long as $\mathbf{y}$ is appreciable,
where $\alpha_{1},\cdots,\alpha_{k}\in\mathbb{DC}$.

###### Proof.

We have

$A\mathbf{y}=A\sum_{j=1}^{k}\mathbf{x}^{(j)}\alpha_{j}=\sum_{j=1}^{k}A\mathbf{x}^{(j)}\alpha_{j}=\sum_{j=1}^{k}\mathbf{x}^{(j)}\lambda\alpha_{j}=\sum_{j=1}^{k}\mathbf{x}^{(j)}\alpha_{j}\lambda=\mathbf{y}\lambda.$

Hence, $\mathbf{y}$ is a right eigenvector of $A$, associated with $\lambda$,
as long as $\mathbf{y}$ is appreciable. Note that by Proposition 4.2,
$\lambda$ is a real number. Thus, we have
$\lambda\alpha_{j}=\alpha_{j}\lambda$. ∎

Suppose that $A\in{\mathbb{DC}}^{n\times n}$ is a Hermitian matrix and has a
real right eigenvalue $\lambda$. Denote the set of right eigenvectors of $A$,
associated with $\lambda$ by $V_{A}(\lambda)$.

The right eigenvectors of a dual complex Hermitian matrix have properties
similar to eigenvectors of a complex Hermitian matrix.

######  Proposition 4.5.

Two right eigenvectors of a Hermitian matrix $A\in{\mathbb{DC}}^{n\times n}$,
associated with two distinct right eigenvalues, are orthogonal to each other.

###### Proof.

Suppose $\mathbf{x}$ and $\mathbf{y}$ are two right eigenvectors of a
Hermitian matrix $A\in{\mathbb{DC}}^{n\times n}$, associated with two distinct
right eigenvalues $\lambda$ and $\mu$, respectively. By Proposition 4.2,
$\lambda$ and $\mu$ are real numbers. Then $\lambda\not=\mu$. We have

$\lambda(\mathbf{x}^{*}\mathbf{y})=(\mathbf{x}\lambda)^{*}\mathbf{y}=(A\mathbf{x})^{*}\mathbf{y}=\mathbf{x}^{*}A\mathbf{y}=\mathbf{x}^{*}\mathbf{y}\mu=\mu\mathbf{x}^{*}\mathbf{y}.$

Since $\lambda\not=\mu$, we have $\mathbf{x}^{*}\mathbf{y}=0$. ∎

Suppose that $A\in{\mathbb{DC}}^{n\times n}$ is a Hermitian matrix, and
$\lambda$ is an eigenvalue of $A_{st}$ with multiplicity $p$. Then $\lambda$
is a real number. If $\lambda$ is also a right eigenvalue of $A$, and there
are right eigenvectors $\\{\mathbf{u}^{(1)},\cdots,\mathbf{u}^{(p)}\\}$ of $A$
associated with $\lambda$ such that
$\\{\mathbf{u}^{(1)},\cdots,\mathbf{u}^{(p)}\\}$ forms an orthonormal basis of
$V_{A}(\lambda)$, then we say that $\lambda$ is a regular eigenvalue of
$A_{st}$. By Theorem 4.3, any single multiple eigenvalue of $A_{st}$ is
regular. If all the eigenvalues of $A_{st}$ are regular, then we say that the
Hermitian matrix $A$ is regular, i.e., $A$ has $n$ right eigenvalues, counting
with multiplicity. Otherwise, we say that $A$ is irregular.

######  Theorem 4.6.

Suppose that $A\in{\mathbb{DC}}^{n\times n}$ is a regular Hermitian matrix.
Then there are unitary matrix $U\in{\mathbb{DC}}^{n\times n}$ and real
diagonal matrix $D\in{\mathbb{R}}^{n\times n}$ such that $D=U^{-1}AU$. The $n$
diagonal entries of $D$ are right eigenvalues of $A$, and the column vectors
of $U$ are corresponding right eigenvectors.

###### Proof.

By above discussion, ${\mathbb{DC}}^{n\times 1}$ has an orthonormal basis such
that the basis vectors are right eigenvectors of $A$. Let them be the column
vectors of $U$. Then we have the desired result. ∎

##  5 Right Subeigenvalues and Subeigenvectors of Hermitian Matrices

Suppose that $A\in{\mathbb{DC}}^{n\times n}$ is a Hermitian matrix. If $A$ is
irregular, then $A$ has no $n$ right eigenvalues, and $A$ cannot be
diagonalized. We also know that the right eigenvalues of $A$ must be
eigenvalues of $A_{st}$, but not vice versa. Where are those “missing”
eigenvalues of $A_{st}$? In this section, we will find out those “missing”
eigenvalues of $A_{st}$ and their roles in the spectral theory of dual complex
Hermitian matrices. We call them right subeigenvalues of $A$.

Suppose that $A\in{\mathbb{DC}}^{n\times n}$. Assume that there is
$\lambda\in{\mathbb{DC}}$, $\mu\in{\mathbb{C}}$, $\mu\not=0$, and
$\mathbf{x},\mathbf{y}\in{\mathbb{DC}}^{n\times 1}$ such that

$A\mathbf{x}=\mathbf{x}\lambda+\mathbf{y}\mu\epsilon\mathbf{j},$ (9)

and

$A\mathbf{y}=\mathbf{y}\lambda-\mathbf{x}\mu\epsilon\mathbf{j},$ (10)

where $\mathbf{x}$ and $\mathbf{y}$ are appreciable, and orthogonal to each
other. Then we say that $\lambda$ is a right subeigenvalue of $A$, with
multiplicity $2$, $\mathbf{x}$ and $\mathbf{y}$ are right subeigenvectors of
$A$, associated with $\lambda$, and $\mu$ is the adjoint parameter of $A$,
associated with $\lambda$.

Here, we require $\mu$ to be a complex number, as if $\mu$ is a dual complex
number, its infinitesimal part does not play any role here. We have the
following proposition parallel to Proposition 4.2.

######  Proposition 5.1.

A right subeigenvalue $\lambda$ of a Hermitian matrix
$A=A_{st}+A_{\mathcal{I}}\epsilon\mathbf{j}\in{\mathbb{DC}}^{n\times n}$ must
be a real number and is a multiple eigenvalue of the complex Hermitian matrix
$A_{st}$.

A right subeigenvalue of a positive semi-definite Hermitian matrix
$A\in{\mathbb{DC}}^{n\times n}$ must be a nonnegative number. A right
subeigenvalue of a positive definite Hermitian matrix
$A\in{\mathbb{DC}}^{n\times n}$ must be a positive number.

###### Proof.

Suppose that $A\in{\mathbb{DC}}^{n\times n}$ is a Hermitian matrix, $\lambda$
is a right subeigenvalue of $A$ with right eigenvectors $\mathbf{x}$ and
$\mathbf{y}$, and the adjoint parameter $\mu$. By (9) and
$\mathbf{x}^{*}\mathbf{y}=0$, we have

$\mathbf{x}^{*}A\mathbf{x}=\mathbf{x}^{*}\mathbf{x}\lambda=\|\mathbf{x}\|^{2}\lambda$

is a real number. Thus, $\lambda$ is a real number. With
$A=A_{st}+A_{\mathcal{I}}\epsilon\mathbf{j}$,
$\mathbf{x}=\mathbf{x}_{st}+\mathbf{x}_{\mathcal{I}}\epsilon\mathbf{j}$ and
$\mathbf{y}=\mathbf{y}_{st}+\mathbf{y}_{\mathcal{I}}\epsilon\mathbf{j}$, from
(9) and (10), we have

$A_{st}\mathbf{x}_{st}=\mathbf{x}_{st}\lambda=\lambda\mathbf{x}_{st},\ \
A_{st}\mathbf{y}_{st}=\mathbf{y}_{st}\lambda=\lambda\mathbf{y}_{st}.$

Since $\mathbf{x}$ and $\mathbf{y}$ are appreciable,
$\mathbf{x}_{st}\not=\mathbf{0}$ and $\mathbf{y}_{st}\not=\mathbf{0}$. By
$\mathbf{x}^{*}\mathbf{y}=0$, we have $\mathbf{x}^{*}_{st}\mathbf{y}_{st}=0$.
Thus, $\mathbf{x}_{st}$ and $\mathbf{y}_{st}$ are two orthogonal eigenvectors
of $A_{st}$, associated with $\lambda$. Then $\lambda$ is a multiple
eigenvalue of $A_{st}$.

The other conclusions can be proved similarly. ∎

We will see that “missing” eigenvalues of $A_{st}$ are recovered as right
subeigenvalues of $A$, if they are not right eigenvalues of $A$. Consider the
case that $\lambda$ is a double eigenvalue of $A_{st}$. This is the case of
Example 2.

######  Proposition 5.2.

Suppose that
$A=A_{st}+A_{\mathcal{I}}\epsilon\mathbf{j}\in{\mathbb{DC}}^{n\times n}$ is a
Hermitian matrix, and $\lambda$ is a double eigenvalue of $A_{st}$. Let
$\\{\mathbf{x}_{st},\mathbf{y}_{st}\\}$ be an orthonormal basis of the
eigenspace of $\lambda$ with respect to $A_{st}$, and

$\mu=\mathbf{y}_{st}^{*}A_{\mathcal{I}}\bar{\mathbf{x}}_{st}.$ (11)

Then $\lambda$ is a double right eigenvalue of $A$ if and only if $\mu=0$.
Otherwise, $\lambda$ is a double right subeigenvalue of $A$, with $\mu$ as its
adjoint parameter. This fact is independent of the choice of the orthonormal
basis $\\{\mathbf{x}_{st},\mathbf{y}_{st}\\}$.

###### Proof.

Since $A_{st}\mathbf{x}_{st}=\lambda\mathbf{x}_{st}$, by some algebraic
derivation, we see that (9) is equivalent to

$(\lambda
I_{n}-A_{st})\mathbf{x}_{\mathcal{I}}=A_{\mathcal{I}}\bar{\mathbf{x}}_{st}-\mu\mathbf{y}_{st}.$
(12)

Similarly, (10) is equivalent to

$(\lambda
I_{n}-A_{st})\mathbf{y}_{\mathcal{I}}=A_{\mathcal{I}}\bar{\mathbf{y}}_{st}+\mu\mathbf{x}_{st}.$
(13)

By matrix theory, (12) and (13) have solutions $\mathbf{x}_{\mathcal{I}}$ and
$\mathbf{y}_{\mathcal{I}}$ if and only if

$\mathbf{x}_{st}^{*}(A_{\mathcal{I}}\bar{\mathbf{x}}_{st}-\mu\mathbf{y}_{st})=0,\
\
\mathbf{y}_{st}^{*}(A_{\mathcal{I}}\bar{\mathbf{x}}_{st}-\mu\mathbf{y}_{st})=0,$

and

$\mathbf{x}_{st}^{*}(A_{\mathcal{I}}\bar{\mathbf{y}}_{st}+\mu\mathbf{x}_{st})=0,\
\
\mathbf{y}_{st}^{*}(A_{\mathcal{I}}\bar{\mathbf{y}}_{st}+\mu\mathbf{x}_{st})=0.$

We may prove these by using (11), the orthogonality between $\mathbf{x}_{st}$
and $\mathbf{y}_{st}$, and the skew-symmetry of $A_{\mathcal{I}}$. Then, by
(9) and (10), $\lambda$ is a double right eigenvalue or subeigenvalue of $A$,
depending upon $\mu=0$ or not. Finally, we may prove the fact that $\mu=0$ or
not is independent from the choice of the orthonormal basis
$\\{\mathbf{x}_{st},\mathbf{y}_{st}\\}$, by choosing another orthonormal basis
to show this, via the skew-symmetry of $A_{\mathcal{I}}$. ∎

When the multiplicity of an eigenvalue of $A_{st}$ is higher than $2$, the
situation is more complicated. Viewing Theorem 4.6, Proposition 5.1, and
Theorem 3.1 of [11], we have the following theorem. In the proof of this
theorem, the symbols $I_{i}$ and $I$ are different from the rest of this
paper.

######  Theorem 5.3.

Suppose that $A\in{\mathbb{DC}}^{n\times n}$ is a Hermitian matrix. Then there
are unitary matrix $U\in{\mathbb{DC}}^{n\times n}$ and block-diagonal matrix
$\Sigma\in{\mathbb{DC}}^{n\times n}$ such that $\Sigma=U^{-1}AU$, and each
block of $\Sigma$ is either of the form:

  * •

$(\lambda_{i})$,

  * •

$\begin{bmatrix}\lambda_{i}&\mu_{i}\epsilon\mathbf{j}\\\
-\mu_{i}\epsilon\mathbf{j}&\lambda_{i}\end{bmatrix}$,

where each $\lambda_{i}$ is real, $\mu_{i}$ is complex, and $\mu_{i}\not=0$.

###### Proof.

Let $A\in{\mathbb{DC}}^{n\times n}$ be a Hermitian matrix. Denote
$A=A_{st}+A_{\mathcal{I}}\epsilon\mathbf{j}$, where
$A_{st},A_{\mathcal{I}}\in{\mathbb{C}}^{n\times n}$. Then $A_{st}$ is
Hermitain, and $A_{\mathcal{I}}$ is skew-symmetric. This implies that there is
a complex unitary matrix $S\in{\mathbb{C}}^{n\times n}$ and a real diagonal
matrix $D\in{\mathbb{R}}^{n\times n}$ such that $D=SA_{st}S^{*}$. Suppose that
$D={\rm diag}(\lambda_{1}I_{1},\lambda_{2}I_{2},\cdots,\lambda_{r}I_{r})$,
where $\lambda_{1}>\lambda_{2}>\cdots>\lambda_{r}$, and $I_{1},\cdots,I_{r}$
are identity matrices such that the sum of their dimensions is $n$. Let
$M=SAS^{*}$. Then

$\displaystyle M$ $\displaystyle=$ $\displaystyle
D+SA_{\mathcal{I}}\epsilon\mathbf{j}S^{*}=D+SA_{\mathcal{I}}S^{\top}\epsilon\mathbf{j}$
$\displaystyle=$ $\displaystyle\begin{bmatrix}\lambda_{1}I_{1}+\epsilon
C_{11}\mathbf{j}&\epsilon C_{12}\mathbf{j}&\cdots&\epsilon C_{1r}\mathbf{j}\\\
-\epsilon C_{12}^{\top}\mathbf{j}&\lambda_{2}I_{2}+\epsilon
C_{22}\mathbf{j}&\cdots&\epsilon C_{2r}\mathbf{j}\\\
\vdots&\vdots&\ddots&\vdots\\\ -\epsilon C_{1r}^{\top}\mathbf{j}&-\epsilon
C_{2r}^{\top}\mathbf{j}&\cdots&\lambda_{r}I_{r}+\epsilon
C_{rr}\mathbf{j}\end{bmatrix},$

where each $C_{ij}$ is a complex matrix of adequate dimensions, each $C_{ii}$
is skew-symmetric.

Let

$P=\begin{bmatrix}I_{1}&{\epsilon
C_{12}\mathbf{j}\over\lambda_{1}-\lambda_{2}}&\cdots&{\epsilon
C_{1r}\mathbf{j}\over\lambda_{1}-\lambda_{r}}\\\ {\epsilon
C_{12}^{\top}\mathbf{j}\over\lambda_{1}-\lambda_{2}}&I_{2}&\cdots&{\epsilon
C_{2r}\mathbf{j}\over\lambda_{2}-\lambda_{r}}\\\
\vdots&\vdots&\ddots&\vdots\\\ {\epsilon
C_{1r}^{\top}\mathbf{j}\over\lambda_{1}-\lambda_{r}}&{\epsilon
C_{2r}^{\top}\mathbf{j}\over\lambda_{2}-\lambda_{r}}&\cdots&I_{r}\end{bmatrix},$

and $N=PMP^{*}\equiv(PS)A(PS)^{*}$. Then

$P^{*}=\begin{bmatrix}I_{1}&-{\epsilon
C_{12}\mathbf{j}\over\lambda_{1}-\lambda_{2}}&\cdots&-{\epsilon
C_{1r}\mathbf{j}\over\lambda_{1}-\lambda_{r}}\\\ -{\epsilon
C_{12}^{\top}\mathbf{j}\over\lambda_{1}-\lambda_{2}}&I_{2}&\cdots&-{\epsilon
C_{2r}\mathbf{j}\over\lambda_{2}-\lambda_{r}}\\\
\vdots&\vdots&\ddots&\vdots\\\ -{\epsilon
C_{1r}^{\top}\mathbf{j}\over\lambda_{1}-\lambda_{r}}&-{\epsilon
C_{2r}^{\top}\mathbf{j}\over\lambda_{2}-\lambda_{r}}&\cdots&I_{r}\end{bmatrix},$

$PP^{*}=P^{*}P=I$, where $I$ is the $n\times n$ identity matrix. Thus, $P$ and
$PS$ are unitary matrices. Also, $N$ is a block diagonal matrix, $N={\rm
diag}(\lambda_{1}I_{1}+\epsilon C_{11}\mathbf{j},\lambda_{2}I_{2}+\epsilon
C_{22}\mathbf{j},\cdots,\lambda_{r}I_{r}+\epsilon C_{rr}\mathbf{j})$. We then
may use the spectral theorem for skew-symmetric matrices [12, 20] to
diagonalize $N$ further to the desired form. ∎

With Theorem 4.6, Proposition 5.1 and Theorem 5.3, we have the following
theorem.

######  Theorem 5.4.

Suppose that $A\in{\mathbb{DC}}^{n\times n}$ is Hermitian. Then $A$ has
exactly $n$ right eigenvalues and subeigenvalues, which are all real numbers.
There are also $n$ right eigenvectors and subeigenvectors, associated with
these $n$ right eigenvalues and subeigenvalues, such that they form an
orthonormal basis of ${\mathbb{DC}}^{n\times 1}$. The Hermitian matrix $A$ is
positive semi-definite or definite if and only all of these right eigenvalues
and subeigenvalues are nonnegative or positive, respectively.

###### Proof.

In the decomposition $\Sigma=U^{-1}AU$ in Theorem 5.3, we see that
$\lambda_{i}$ in the block $(\lambda_{i})$ is a right eigenvalue of $A$, while
$\lambda_{i}$ in the block

$\begin{bmatrix}\lambda_{i}&\mu_{i}\epsilon\mathbf{j}\\\
-\mu_{i}\epsilon\mathbf{j}&\lambda_{i}\end{bmatrix}$

is a right subeigenvalue of $A$, with multiplicity $2$, and the columns of $U$
are corresponding right eigenvectors and subeigenvectors. The other
conclusions follow from Propositions 4.2 and 5.1. ∎

The following corollary complements Theorem 4.3 and Proposition 5.2.

######  Corollary 5.5.

Suppose that
$A=A_{st}+A_{\mathcal{I}}\epsilon\mathbf{j}\in{\mathbb{DC}}^{n\times n}$ is
Hermitian, and $\lambda$ is an eigenvalue of $A_{st}$ with multiplicity $p$.
Then there is an integer $k$ satisfying $0\leq 2k\leq p$, such that $\lambda$
is a right eigenvalue of $A$, with multiplicity $p-2k$, if $p-2k>0$, and a
right subeigenvalue of $A$, with multiplicity $2k$, if $k>0$.

##  6 Singular Value Decomposition of Dual Complex Matrices

Suppose that $A\in{\mathbb{DC}}^{m\times n}$. Then
$A^{*}A\in{\mathbb{DC}}^{n\times n}$ is a positive semi-definite Hermitian
matrix. By Theorem 5.4, $A^{*}A$ has exactly $n$ right eigenvalues and
subeigenvalues, and they are all nonnegative numbers. Assume that $A^{*}A$ has
$r$ positive right eigenvalues and subeigenvalues. Then we say that the
standard rank of $A$ is $r$.

We now present the singular value decomposition of dual complex matrices. The
following theorem generalizes Theorem 4.4 of [11] for square dual number
matrices to general dual complex matrices.

######  Theorem 6.1.

Suppose that $A\in{\mathbb{DC}}^{m\times n}$ with standard rank $r$. Then
there exist a dual complex unitary matrix $U\in{\mathbb{DC}}^{m\times m}$ and
a dual complex unitary matrix $V\in{\mathbb{DC}}^{n\times n}$ such that

$U^{*}AV=\begin{bmatrix}\Sigma_{r}&O&O\\\ O&D\epsilon\mathbf{j}&O\\\
O&O&O\end{bmatrix},$ (14)

where $\Sigma_{r}\in{\mathbb{DC}}^{r\times r}$ is a block-diagonal matrix, and
each block of $\Sigma_{r}$ is either of the form:

  * •

$(\sigma_{i})$,

  * •

$\begin{bmatrix}\sigma_{i}&\nu_{i}\epsilon\mathbf{j}\\\
-\nu_{i}\epsilon\mathbf{j}&\sigma_{i}\end{bmatrix}$,

each $\sigma_{i}$ is real and positive, $\nu_{i}$ is complex, $\nu_{i}\not=0$,
and $D$ is a $p\times p$ positive diagonal matrix, $r+p\leq l=\min\\{m,n\\}$.

###### Proof.

Suppose that positive semi-definite Hermitian matrix $A^{*}A$ has right
eigenvalues and subeigenvalues $\sigma_{1}^{2},\cdots,\sigma_{n}^{2}$,
satisfying $\sigma_{1}\geq\cdots\geq\sigma_{r}>0$,
$\sigma_{r+1}=\cdots=\sigma_{n}=0$. By Theorem 5.4, there are orthogonal right
eigenvectors and subeigenvectors $\mathbf{v}^{(1)},\cdots,\mathbf{v}^{(n)}$ of
$A^{*}A$, associated with $\sigma_{1}^{2},\cdots,\sigma_{n}^{2}$,
respectively. Write
$V^{\prime}_{1}=(\mathbf{v}^{(1)},\cdots,\mathbf{v}^{(r)})$,
$V^{\prime}_{2}=(\mathbf{v}^{(r+1)},\cdots,\mathbf{v}^{(n)})$,
$V^{\prime}=(V^{\prime}_{1},V^{\prime}_{2})$. Let $\Sigma_{r}$ be block-
diagonal matrix described in the theorem. Then we have

$A^{*}AV^{\prime}_{1}=V^{\prime}_{1}\Sigma_{r}^{2},$
$(V^{\prime}_{1})^{*}A^{*}AV^{\prime}_{1}=\Sigma_{r}^{2},$
$A^{*}AV^{\prime}_{2}=O,$ $(V^{\prime}_{2})^{*}A^{*}AV^{\prime}_{2}=O.$

Therefore, $AV^{\prime}_{2}=B\epsilon\mathbf{j}$ for a complex matrix $B$. Let
$U^{\prime}_{1}=AV^{\prime}_{1}\Sigma_{r}^{-1}$. Then

$(U^{\prime}_{1})^{*}AV^{\prime}_{2}=\left(\Sigma_{r}^{-1}\right)^{*}(V^{\prime}_{1})^{*}(A^{*}AV^{\prime}_{2})=\left(\Sigma_{r}^{-1}\right)^{*}(V^{\prime}_{1})^{*}O=O.$

Also observe that $(U^{\prime}_{1})^{*}U^{\prime}_{1}=I_{r}$. Take
$U^{\prime}_{2}\in{\mathbb{DC}}^{m\times(m-r)}$ such that
$U^{\prime}=(U^{\prime}_{1},U^{\prime}_{2})$ is a unitary matrix. We see that

$(U^{\prime}_{2})^{*}B\epsilon\mathbf{j}=G\epsilon\mathbf{j},$

where $G$ is an $(m-r)\times(n-r)$ complex matrix. Then

$\displaystyle(U^{\prime})^{*}AV^{\prime}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}(U^{\prime}_{1})^{*}AV^{\prime}_{1}&(U^{\prime}_{1})^{*}AV^{\prime}_{2}\\\
(U^{\prime}_{2})^{*}AV^{\prime}_{1}&(U^{\prime}_{2})^{*}AV^{\prime}_{2}\end{bmatrix}$
$\displaystyle=$
$\displaystyle\begin{bmatrix}(U^{\prime}_{1})^{*}U^{\prime}_{1}\Sigma_{r}&O\\\
(U^{\prime}_{2})^{*}U^{\prime}_{1}\Sigma_{r}&G\epsilon\mathbf{j}\end{bmatrix}$
$\displaystyle=$ $\displaystyle\begin{bmatrix}\Sigma_{r}&O\\\
O&G\epsilon\mathbf{j}\end{bmatrix}.$

After taking SVD decomposition of the complex matrix $G$, we have the desired
result. ∎

We call the decomposition (14) the singular value decomposition (SVD) of the
matrix $A$. Assume that $D={\rm diag}(\sigma_{r+1},\cdots,\sigma_{r+p})$,
where $\sigma_{r+1}\geq\cdots\geq\sigma_{r+p}>0$, and $r+p\leq
l=\min\\{m,n\\}$. We call $\sigma_{1},\cdots,\sigma_{r}$ the standard singular
values of $A$, $\sigma_{r+1},\cdots,\sigma_{r+p}$ the infinitesimal singular
values of $A$, and $\sigma_{r+p+1}=\cdots=\sigma_{l}=0$ the zero singular
values of $A$, and $p$ the infinitesimal rank of $A$.

##  7 Extensions to Dual Quaternion Matrices

In this section, we briefly study extensions of our results to dual quaternion
matrices.

We denote the quaternions and the dual quaternions by ${\mathbb{Q}}$ and
$\mathbb{DQ}$, respectively. A dual quaternion $q$ has the form

$q=q_{st}+q_{\mathcal{I}}\epsilon,$

where $q_{st},q_{\mathcal{I}}\in{\mathbb{Q}}$ are the standard part and the
infinitesimal part of $q$ respectively,

$q_{st}=q_{0}+q_{1}\mathbf{i}+q_{2}\mathbf{j}+q_{3}\mathbf{k},$
$q_{\mathcal{I}}=q_{4}+q_{5}\mathbf{i}+q_{6}\mathbf{j}+q_{7}\mathbf{k},$

$q_{0},q_{1},q_{2},q_{3},q_{4},q_{5},q_{6}$ and $q_{7}$ are real numbers. The
multiplication of dual quaternions is also noncommutative.

The conjugate of $q=q_{st}+q_{\mathcal{I}}\epsilon$ is

$\bar{q}=\bar{q}_{st}-q_{\mathcal{I}}\epsilon,$

with

$\bar{q}_{st}=q_{0}-q_{1}\mathbf{i}-q_{2}\mathbf{j}-q_{3}\mathbf{k}.$

The magnitude of $q$ is

$|q|=|q_{st}|=\sqrt{q_{0}^{2}+q_{1}^{2}+q_{2}^{2}+q_{3}^{2}}.$

The collections of quaternion and dual quaternion $m\times n$ matrices are
denoted by ${\mathbb{Q}}^{m\times n}$ and ${\mathbb{DQ}}^{m\times n}$,
respectively.

A dual quaternion matrix $A=(a_{ij})\in{\mathbb{DQ}}^{m\times n}$ can be
denoted as

$A=A_{st}+A_{\mathcal{I}}\epsilon,$ (15)

where $A_{st},A_{\mathcal{I}}\in{\mathbb{Q}}^{m\times n}$. The transpose of
$A$ is $A^{\top}=(a_{ji})$. The conjugate of $A$ is $\bar{A}=(\bar{a}_{ij})$.
The conjugate transpose of $A$ is $A^{*}=(\bar{a}_{ji})=\bar{A}^{\top}$.

With these settings, in the extensions of our results to dual quaternion
matrices, $\epsilon\mathbf{j}$ appeared in many places of Sections 2-6 of this
paper may be changed to $\epsilon$ only.

There are two major differences between dual complex matrices and dual
quaternion matrices.

The first difference is that the standard parts and the infinitesimal parts of
dual complex numbers are complex numbers. The multiplication between complex
numbers is commutative. What we need to take care in Sections 2-6 is the
multiplication between $\mathbf{j}$ and complex numbers. Thus, we need to use
Lemma 3.1. The standard parts and the infinitesimal parts of dual quaternions
are quaternions. The multiplication between quaternions is not commutative.
Thus, additional care is needed in the extensions. For example, (4) only can
be written as

$\mathbf{x}_{st}\lambda_{\mathcal{I}}=A_{\mathcal{I}}\mathbf{x}_{st}+A_{st}\mathbf{x}_{\mathcal{I}}-\mathbf{x}_{\mathcal{I}}\lambda_{st}.$
(16)

The second difference is that the theory of right eigenvalues and
subeigenvalues of dual complex matrices is based upon eigenvalues of complex
matrices, while the theory of right eigenvalues and subeigenvalues of dual
quaternion matrices needs to be based upon right eigenvalues of quaternion
matrices [18, 21]. An $n\times n$ complex matrix has $n$ eigenvalues, while an
$n\times n$ quaternion matrix has $2n$ right eigenvalues [18, 21].

These two differences make the extension of Section 3 to dual quaternion
matrices needs additional care.

Coming to Sections 4 and 5 of this paper, the extensions are slightly simpler,
as the right eigenvalues and subeigenvalues of Hermitian matrices are real
numbers, while multiplications between a real number and a dual quaternion is
commutative.

Then, for Theorem 5.3 of this paper, a spectral theorem for skew-symmetric
quaternion matrices is needed. A reference for this is [16].

##  8 Final Remarks

In this paper, we studied right eigenvalues, right subeigenvalues and singular
value decomposition of dual complex matrices. By introducing right
subeigenvalues, we fully characterize the spectral theory of dual complex
Hermitian matrices. An $n\times n$ dual complex Hermitian matrix has exactly
$n$ right eigenvalues and subeigenvalues, which are all real. The Hermitian
matrix is positive semi-definite or definite if and only if all of its right
eigenvalues and subeigenvalues are nonnegative or positive, respectively.
Based upon these, we present the singular value decomposition for general dual
complex matrices. We expect that these are also true for dual quaternion
matrices. We also think that these may be very useful in the future
applications. Further study on this subject would be fruitful and useful.

Acknowledgment We are thankful to Prof. Hong Yan for showing us dual numbers,
dual quaternions and geometric algebra, and their applications in robotics and
computer graphics. We are grateful to Ran Gutin who pointed out an error and
made detailed comments on Section 6 of the early version of this paper, and to
Chen Ling who read our draft carefully.

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