The Hermitian Solution to a New System of Commutative Quaternion Matrix Equations

Preprint

Article

The Hermitian Solution to a New System of Commutative Quaternion Matrix Equations

Altmetrics

Downloads

Views

Comments

A peer-reviewed article of this preprint also exists.

Yue Zhang,

Qing-Wen Wang^*,Lv-Ming Xie

Yue Zhang,

Qing-Wen Wang^*,Lv-Ming Xie

This version is not peer-reviewed

Submitted:

21 February 2024

Posted:

22 February 2024

You are already at the latest version

Alerts

Abstract

This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression when it is solvable. To illustrate our main findings, we present two numerical algorithms and examples in this paper.

Keywords:

Subject: Physical Sciences - Mathematical Physics

MSC: 15A09; 15A24; 15B33; 15B57

1. Introduction

In 1843, Hamilton introduced the concept of real quaternions which are defined by [1]

H = {q = q_{0} + q_{1} i + q_{2} j + q_{3} k : i^{2} = j^{2} = k^{2} = - 1, i j k = - 1, q_{0}, q_{1}, q_{2}, q_{3} \in R},

which is a four-dimensional noncommutative associative algebra over real number field. Quaternions have been used in many areas such as statistic of quaternion random signals [2], color image processing [3] and face recognition [4]. Real quaternions are an extension of the complex numbers. However, the multiplication of the real quaternions is non-commutative which causes many difficulties.

A commutative quaternion, which was introduced by Segre [5] in 1892, is in the form of

q = q_{0} + q_{1} i + q_{2} j + q_{3} k

, where

q_{0}, q_{1}, q_{2}, q_{3}

belong to the real number field, and the imaginary identities

i, j, k

satisfy

i^{2} = k^{2} = - 1, j^{2} = 1, i j k = - 1, i j = j i = k, j k = k j = i, k i = i k = - j

. The most prominent feature of a commutative quaternion is the satisfaction of the multiplication commutative rule. The collection of commutative quaternions comprises a four-dimensional Clifford algebra, forming a ring. Within this set, we can find noteworthy attributes such as nontrivial idempotents, zero divisors, and nilpotent elements. There are many applications of the commutative quaternion algebra in Hopfield neural netwoks, digital signal, image processing [6,7,8,9,10], and so on. Commutative quaternions are also extensively researched. K

\ddot{o}

sal et al. [11] gave complex representations of commutative quaternion matrices and discussed several related properties. In [12], K

\ddot{o}

sal et al. proposed the real representation of a commutative quaternion matrix, and derived some explicit expression of the solutions of the commutative quaternion matrix equations

X - A \bar{\bar{X}} B = C, X - A \bar{\tilde{X}} B = C

and

X - A \tilde{\bar{X}} B = C

, which are called the Kalman-Yakubovich-conjugate matrix equations, by means of real representation of a commutative quaternion matrix. Based on this, K

\ddot{o}

sal et al. [13] gave an expression of the general solution to the matrix equation

A X = B

over the commutative quaternion ring.

Hermitian matrix has drawn a lot of attentions due to its great importance. In [14], Yu et al. studied Hermitian solutions to the generalizaed quaternion matrix equation

A X B + C X^{*} D = E

through the real representation method. Yuan et al. [15] discussed Hermitian solutions to the split quaternion matrix equation

A X B + C X D = E

by using the complex representation method. In [24], Kyrchei obtained the determinantal representation formulas of

η

η

-skew)-Hermitian solutions to the quaternion matrix equations

A X = B

and

A X A^{η *} = B

. As far as we know that the Sylvester matrix equations have a large number of applications in different fields. For example, the Sylvester matrix equation

A_{1} X + X B_{1} = C_{1}

and the Sylvester-like matrix equation

A_{1} X + Y B_{1} = C_{1}

have been applied in singular system control [16], perturbation theory [17], sensitivity analysis [18] and control theory [19]. Wang et al. [20,21] considered the system of coupled Sylvester-like quaternion matrix equations. In [22], Wang et al. derived solvability conditions and expressions of the general solution to the system of two-sides coupled Sylvester-like quaternion matrix equations. Kyrchei [23] gave the determinantal representation formulas of solutions to the generalized Sylvester quaternion matrix equation

A_{1} X_{1} B_{1} + A_{2} X_{2} B_{2} = C

Motivated by keeping interests in Hermitian solutions and applications of the system of commutative quaternion matrix equations, we in this paper intend to investigate the solvability conditions and the Hermitian solutions to the following system of commutative quaternion matrix equations

\{\begin{matrix} A_{1} X = C_{1}, \\ Y B_{1} = D_{1}, \\ A_{2} Z = C_{2}, Z B_{2} = D_{2}, \\ A_{3} W = C_{3}, W B_{3} = D_{3}, A_{4} W B_{4} = C_{4}, \\ A_{5} X + Y B_{5} + A_{6} Z B_{6} + A_{7} W B_{7} = C_{5}, \end{matrix}

(1)

where

X, Y, Z, W

are unknown Hermitian commutative quaternion matrices.

This paper is organized as follows. In Section 2, we review some useful properties and the structures of

vec (A X B)

over the commutative quaternion algebra when X is a Hermitian commutative quaternion matrix. In Section 3, we derive some practical necessary and sufficient conditions for the existence of Hermitian solutions to the system (1) over

H_{c}

, and the numerical examples are given in Section 4.

2. Preliminaries

Throughout this paper, let

R^{m \times n}, {SR}^{m \times n}, {ASR}^{m \times n}, C^{m \times n}, H_{c}, {H_{c}}^{n}, {H_{c}}^{m \times n}

be the set of all

m \times n

real matrices, the set of all

n \times n

real symmetric matrices, the set of all

n \times n

real anti-symmetric matrices, the set of all

m \times n

complex matrices, the set of commutative quaternions, the set of n dimensional commutative quaternion column vectors, and the set of all

m \times n

commutative quaternion matrices, respectively. The symbol

r (A)

denotes the rank of A. Let the symbols

I, O, A^{T}, A^{†}

stand for the identity matrix, the zero matrix with appropriate size, the transpose of A, and the Moore-Penrose inverse of a matrix A, respectively.

\bar{A}

and

A^{H}

denote the conjugate matrix, the conjugate transpose matrix of A, respectively. We call

A \in {H_{c}}^{n \times n}

is a Hermitian matrix if

A^{H} = A

, and denote it by

A \in {HH}_{c}^{n \times n}

, where

{HH}_{c}^{n \times n}

is the set of all Hermitian commutative quaternion matrices with the size of

n \times n

For any

A \in {H_{c}}^{m \times n}

, A can be uniquely expressed as

A = A_{0} + A_{1} i + A_{2} j + A_{3} k

, where

A_{0}, A_{1}, A_{2}, A_{3} \in R^{m \times n}

. It can also be uniquely expressed as

A = C_{1} + C_{2} j

, where

C_{1} = A_{0} + A_{1} i, C_{2} = A_{2} + A_{3} i \in C^{m \times n}

Proposition 1.

[11]The complex representation matrix for commutative quaternion

q = d_{1} + d_{2} j, d_{1} = q_{0} + q_{1} i, d_{2} = q_{2} + q_{3} i

is denoted as

g (q) = (\begin{matrix} d_{1} & d_{2} \\ d_{2} & d_{1} \end{matrix}) \in C^{2 \times 2} .

Similarly, for any given

A = A_{1} + A_{2} j \in {H_{c}}^{m \times n}

, the complex representation matrix of A is

G (A) = (\begin{matrix} A_{1} & A_{2} \\ A_{2} & A_{1} \end{matrix}) \in C^{2 m \times 2 n} .

(2)

Obviously,

G (A)

is uniquely determined by A. It is straightforward to confirm that the following statements are valid.

Proposition 2.

[11]If

A, B \in {H_{c}}^{n \times n}

, then

(a): $A = B$ if and only if $G (A) = G (B)$ ,
(b): $G (A + B) = G (A) + G (B)$ ,
(c): $G (I_{n}) = I_{2 n}$ ,
(d): $G (A B) = G (A) G (B)$ .

Suppose

A = (a_{i j}) \in H_{c}^{m \times n}

and

B = (b_{i j}) \in H_{c}^{s \times t}

, the Kronecker product of A and B is defined as

A \otimes B = (a_{i j} B) \in H_{c}^{m s \times n t}

. Considering commutative quaternion matrices

A, B, C, D, E

with appropriate dimensions, along with the real number p, we establish

\begin{matrix} (p A) \otimes B & = A \otimes (p B) = p (A \otimes B), \\ (A, B, C) \otimes D & = (A \otimes D, B \otimes D, C \otimes D), \\ (\begin{matrix} A & B \\ C & D \end{matrix}) \otimes E & = (\begin{matrix} A \otimes E & B \otimes E \\ C \otimes E & D \otimes E \end{matrix}) . \end{matrix}

The vec-operator of

A = (a_{i j}) \in H_{c}^{m \times n}

is defined as

vec (A) = {(a_{1}, a_{2}, \dots, a_{n})}^{T}, a_{j} = (a_{1 j}, a_{2 j}, \dots, a_{m j}), j = 1, 2, \dots, n .

To investigate the Hermitian solutions of a system of matrix equations (1) within the framework of the commutative quaternion algebra, we need to review some certain definitions and fundamental properties.

Assume that

A = A_{1} + A_{2} j \in {H_{c}}^{m \times n}, A_{1}, A_{2} \in C^{m \times n}

, then we have

A_{1} + A_{2} j = A ≅ Φ_{A} = (A_{1}, A_{2}),

where the symbol ≅ represents an equivalence relation. For a given matrix

A = (a_{i j}) \in C^{m \times n}

, the corresponding Frobenius norm is defined as follows:

| | A | | = \sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{n} {∥a_{i j}∥}^{2}}, {∥a_{i j}∥}^{2} = {(Re a_{i j})}^{2} + {(Im a_{i j})}^{2} .

According to the previously mentioned definition of Frobenius norm for complex matrices, we can define the Frobenius norm for commutative quaternion matrix

A = A_{1} + A_{2} j \in {H_{c}}^{m \times n}

as follows:

∥ \overset{´}{A} ∥ = \sqrt{{∥Re A_{1}∥}^{2} + {∥Im A_{1}∥}^{2} + {∥Re A_{2}∥}^{2} + {∥Im A_{2}∥}^{2}},

where

\overset{´}{A} = (\begin{matrix} Re (A_{1}) & Im (A_{1}) & Re (A_{2}) & Im (A_{2}) \end{matrix}),

then we have

| | Φ_{A} | | = | | \overset{´}{A} | | = | | {vec}_{Φ_{A}} | | .

Theorem 1.

[28]Let

p \in R, A, B \in {H_{c}}^{m \times n}

and

C \in {H_{c}}^{n \times s}

. Then

(a): $A = B$ if and only if $Φ_{A} = Φ_{B}$ ,
(b): $Φ_{A + B} = Φ_{A} + Φ_{B}, Φ_{p A} = p Φ_{A}$ ,
(c): ${(A C)}^{T} = C^{T} A^{T}$ ,
(d): ${(A C)}^{- 1} = C^{- 1} A^{- 1}$ , if the matrices $A, C$ and $A C$ are invertible,
(e): $Φ_{A C} = Φ_{A} G (C)$ .

For the purpose of deriving the Hermitian solutions of the system (1), we introduce some relevant definitions and conclusions.

Definition 1.

[15]For the matrix

A = (a_{i j}) \in H_{c}^{n \times n}

, set

a_{1} = (a_{11}, \sqrt{2} a_{21}, \dots, \sqrt{2} a_{n 1}), a_{2} =

(a_{22}, \sqrt{2} a_{32}, \dots, \sqrt{2} a_{n 2}), \dots, a_{n - 1} = (a_{(n - 1) (n - 1)}, \sqrt{2} a_{n (n - 1)}), a_{n} = a_{n n}

, and denote by

{vec}_{S} (A)

the following vector:

{vec}_{S} (A) = {(a_{1}, a_{2}, \dots, a_{n - 1}, a_{n})}^{T} \in H_{c}^{(n (n + 1)) / 2} .

(3)

Definition 2.

[15]For the matrix

B = (b_{i j}) \in H_{c}^{n \times n}

, set

b_{1} = (b_{21}, b_{31}, \dots, b_{n 1}), b_{2} = (b_{32}, b_{42}

\dots, b_{n 2}), \dots, b_{n - 2} = (b_{(n - 1) (n - 2)}, b_{n (n - 2)}), b_{n - 1} = b_{n (n - 1)}

, and denote by

{vec}_{A} (B)

the following vector:

{vec}_{A} (B) = \sqrt{2} {(b_{1}, b_{2}, \dots, b_{n - 2}, b_{n - 1})}^{T} \in H_{c}^{(n (n - 1)) / 2} .

(4)

Proposition 3.

[25]Suppose that

X \in R^{n \times n}

, then

(1)

X \in {SR}^{n \times n} ⟺ vec (X) = K_{S} {vec}_{S} (X),

(5)

where the matrix

K_{S} \in R^{n^{2} \times (n (n + 1) / 2)}

is of the following form:

K_{S} = \frac{1}{\sqrt{2}} (\begin{matrix} \sqrt{2} e_{1} & e_{2} & \dots & e_{n - 1} & e_{n} & 0 & 0 & \dots & 0 & \dots & 0 & 0 & 0 \\ 0 & e_{1} & \dots & 0 & 0 & \sqrt{2} e_{2} & e_{3} & \dots & e_{n} & \dots & 0 & 0 & 0 \\ 0 & 0 & \dots & 0 & 0 & 0 & e_{2} & \dots & 0 & \dots & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & \dots & e_{1} & 0 & 0 & 0 & \dots & 0 & \dots & \sqrt{2} e_{n - 1} & e_{n} & 0 \\ 0 & 0 & \dots & 0 & e_{1} & 0 & 0 & \dots & e_{2} & \dots & 0 & e_{n - 1} & \sqrt{2} e_{n} \end{matrix}),

and

e_{i}

is the ith column of the identity matrix of order n.

(2)

X \in {ASR}^{n \times n} ⟺ vec (X) = K_{A} {vec}_{A} (X),

(6)

{vec}_{A} (X)

is described as (4) and the matrix

K_{A} \in R^{n^{2} \times (n (n - 1) / 2)}

is of the following form:

K_{A} = \frac{1}{\sqrt{2}} (\begin{matrix} e_{2} & e_{3} & \dots & e_{n - 1} & e_{n} & 0 & \dots & 0 & 0 & \dots & 0 \\ - e_{1} & 0 & \dots & 0 & 0 & e_{3} & \dots & e_{n - 1} & e_{n} & \dots & 0 \\ 0 & - e_{1} & \dots & 0 & 0 & - e_{2} & \dots & 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & 0 & 0 & 0 \\ 0 & 0 & \dots & - e_{1} & 0 & 0 & \dots & - e_{2} & 0 & \dots & e_{n} \\ 0 & 0 & \dots & 0 & - e_{1} & 0 & \dots & 0 & - e_{2} & \dots & - e_{n - 1} \end{matrix}),

where

e_{i}

is the column of the identity matrix of order n. It is apparent that

K_{S}^{T} K_{S} = I_{(n (n + 1)) / 2}, K_{A}^{T} K_{A} = I_{(n (n - 1)) / 2}

Next, we explore the relationships between the Hermitian commutative quaternion matrices and symmetric matrices, as well as anti-symmetric matrices.

X = X_{1} + X_{2} j \in {HH}_{c}^{n \times n}

, where

X_{1}, X_{2} \in C^{n \times n}

, we can get

X \in {HH}_{c}^{n \times n} \Leftrightarrow \{\begin{matrix} Re {(X_{1})}^{T} = Re (X_{1}), Im {(X_{1})}^{T} = - Im (X_{1}), \\ Re {(X_{2})}^{T} = - Re (X_{2}), Im {(X_{2})}^{T} = - Im (X_{2}) . \end{matrix}

Apparently,

Re (X_{1})

is symmetric, and

Im (X_{1}), Re (X_{2}), Im (X_{2})

are antisymmetric. By means of Proposition 3, we have the following:

Theorem 2.

[29]Assume that

X = X_{1} + X_{2} j \in {HH}_{c}^{n \times n}

, then we obtain

(\begin{matrix} vec (X_{1}) \\ vec (X_{2}) \end{matrix}) = M (\begin{matrix} {vec}_{S} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \end{matrix}),

(7)

in which

M = (\begin{matrix} K_{S} & i K_{A} & 0 & 0 \\ 0 & 0 & K_{A} & i K_{A} \end{matrix}) .

(8)

Theorem 3.

[29]Suppose that

A = A_{1} + A_{2} j \in {H_{c}}^{m \times n}, B = B_{1} + B_{2} j \in {H_{c}}^{s \times t}

and

X = X_{1} + X_{2} j \in {H_{c}}^{n \times s}

, where

A_{1}, A_{2} \in C^{m \times n}, B_{1}, B_{2} \in C^{s \times t}

and

X_{1}, X_{2} \in C^{n \times s}

. Then

vec (Φ_{A X B}) = G [(B_{1}^{T} \otimes A_{1} + B_{2}^{T} \otimes A_{2}) + (B_{2}^{T} \otimes A_{1} + B_{1}^{T} \otimes A_{2}) j] (\begin{matrix} vec (X_{1}) \\ vec (X_{2}) \end{matrix}) .

(9)

Note that the results of

vec (Φ_{A X B})

is very important for figuring out the system of commutative quaternion matrix equations (1). Analogous methods and related conclusions can be found in [15]. By incorporating Theorem 3 with Theorem 2, we can gain the following outcome.

Theorem 4.

[29]If

A = A_{1} + A_{2} j \in H_{c}^{m \times n}

X = X_{1} + X_{2} j \in {HH}_{c}^{n \times n}

, and

B = B_{1} + B_{2} j \in H_{c}^{n \times s}

, where

A_{i} \in C^{m \times n}, X_{i} \in C^{n \times n}

, and

B_{i} \in C^{n \times s} (i = 1, 2)

. Consequently,

vec (Φ_{A X B}) = G [(B_{1}^{T} \otimes A_{1} + B_{2}^{T} \otimes A_{2}) + (B_{2}^{T} \otimes A_{1} + B_{1}^{T} \otimes A_{2}) j] M (\begin{matrix} {vec}_{S} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \end{matrix}) .

(10)

Lemma 1.

[26]The matrix equation

A x = b

, with

A \in R^{m \times n}

and

b \in R^{m}

, has a solution

x \in R^{n}

if and only if

A A^{†} b = b,

(11)

In this case, it has the general solution

x = A^{†} b + (I_{n} - A^{†} A) y,

(12)

where

y \in R^{n}

is an arbitrary vector, and it has the unique solution

x = A^{†} b

for the case when

r (A) = n

. The solution of the matrix equation

A x = b

with the least norm is

x = A^{†} b

3. The Hermitian solution to the system

In accordance with the above discussion, now we focus on solving the system (1), for ease of description, We firstly state the following notations.

Let

A_{1} = A_{11} + A_{12} j, A_{2} = A_{21} + A_{22} j, A_{3} = A_{31} + A_{32} j \in H_{c}^{m \times n}, C_{1}, C_{2}, C_{3} \in

H_{c}^{m \times n}, B_{1} = B_{11} + B_{12} j, B_{2} = B_{21} + B_{22} j, B_{3} = B_{31} + B_{32} j \in H_{c}^{n \times k}, D_{1}, D_{2}, D_{3} \in H_{c}^{n \times k}, A_{4} = A_{41} + A_{42} j \in H_{c}^{s \times n}, B_{4} = B_{41} + B_{42} j \in H_{c}^{n \times t}, C_{4} \in H_{c}^{s \in \times t}, A_{5} = A_{51} + A_{52} j, A_{6} = A_{61} + A_{62} j, A_{7} = A_{71} + A_{72} j, B_{5} = B_{51} + B_{52} j, B_{6} = B_{61} + B_{62} j, B_{7} = B_{71} + B_{72} j \in H_{c}^{n \times n}

and

C_{5} \in H_{c}^{n \times n}

. We set

E = (\begin{matrix} G [(I \otimes A_{11}) + (I \otimes A_{12}) j] \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ G [(I \otimes A_{51}) + (I \otimes A_{52}) j] \end{matrix}) M, F = (\begin{matrix} 0 \\ 0 \\ 0 \\ G [(B_{11}^{T} \otimes I) + (B_{12}^{T} \otimes I) j] \\ 0 \\ 0 \\ 0 \\ G [(B_{51}^{T} \otimes I) + (B_{52}^{T} \otimes I) j] \end{matrix}) M,

P = (\begin{matrix} 0 \\ G [(I \otimes A_{21}) + (I \otimes A_{22}) j] \\ 0 \\ 0 \\ G [(B_{21}^{T} \otimes I) + (B_{22}^{T} \otimes I) j] \\ 0 \\ 0 \\ G [(B_{61}^{T} \otimes A_{61} + B_{62}^{T} \otimes A_{62}) + (B_{62}^{T} \otimes A_{61} + B_{61}^{T} \otimes A_{62}) j] \end{matrix}) M,

Q = (\begin{matrix} 0 \\ 0 \\ G [(I \otimes A_{31}) + (I \otimes A_{32}) j] \\ 0 \\ 0 \\ G [(B_{31}^{T} \otimes I) + (B_{32}^{T} \otimes I) j] \\ G [(B_{41}^{T} \otimes A_{41} + B_{42}^{T} \otimes A_{42}) + (B_{42}^{T} \otimes A_{41} + B_{41}^{T} \otimes A_{42}) j] \\ G [(B_{71}^{T} \otimes A_{71} + B_{72}^{T} \otimes A_{72}) + (B_{72}^{T} \otimes A_{71} + B_{71}^{T} \otimes A_{72}) j] \end{matrix}) M,

T = (\begin{matrix} vec (Φ_{C_{1}}) \\ vec (Φ_{C_{2}}) \\ vec (Φ_{C_{3}}) \\ vec (Φ_{D_{1}}) \\ vec (Φ_{D_{2}}) \\ vec (Φ_{D_{3}}) \\ vec (Φ_{C_{4}}) \\ vec (Φ_{C_{5}}) \end{matrix}), T_{1} = (\begin{matrix} vec (Re Φ_{C_{1}}) \\ vec (Re Φ_{C_{2}}) \\ vec (Re Φ_{C_{3}}) \\ vec (Re Φ_{D_{1}}) \\ vec (Re Φ_{D_{2}}) \\ vec (Re Φ_{D_{3}}) \\ vec (Re Φ_{C_{4}}) \\ vec (Re Φ_{C_{5}}) \end{matrix}), T_{2} = (\begin{matrix} vec (Im Φ_{C_{1}}) \\ vec (Im Φ_{C_{2}}) \\ vec (Im Φ_{C_{3}}) \\ vec (Im Φ_{D_{1}}) \\ vec (Im Φ_{D_{2}}) \\ vec (Im Φ_{D_{3}}) \\ vec (Im Φ_{C_{4}}) \\ vec (Im Φ_{C_{5}}) \end{matrix}), ϵ = (\begin{matrix} T_{1} \\ T_{2} \end{matrix}),

W = (\begin{matrix} K_{S} & 0 & 0 & 0 \\ 0 & K_{A} & 0 & 0 \\ 0 & 0 & K_{A} & 0 \\ 0 & 0 & 0 & K_{A} \end{matrix}), W = (\begin{matrix} W & 0 & 0 & 0 \\ 0 & W & 0 & 0 \\ 0 & 0 & W & 0 \\ 0 & 0 & 0 & W \end{matrix}),

vec (\vec{X}) = (\begin{matrix} {vec}_{S} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \end{matrix}), vec (\vec{Y}) = (\begin{matrix} {vec}_{S} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \end{matrix}),

vec (\vec{Z}) = (\begin{matrix} {vec}_{S} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \end{matrix}), vec (\vec{W}) = (\begin{matrix} {vec}_{S} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \end{matrix}),

(13)

and

\begin{matrix} E_{1} & = Re E, E_{2} = Im E, F_{1} = Re F, F_{2} = Im F, \\ P_{1} & = Re P, P_{2} = Im P, Q_{1} = Re Q, Q_{2} = Im Q, \\ U_{1} & = (E_{1}, F_{1}, P_{1}, Q_{1}), \\ U_{2} & = (E_{2}, F_{2}, P_{2}, Q_{2}) . \end{matrix}

(14)

For further study the structure of Hermitian solution of the system of matrix equations (1), it is necessary to study the generalized inverse of matrix in the form of column blocks. The following notations are required. Let

\begin{matrix} d = 6 m n + 6 k n + 2 s t + 2 n^{2}, \\ H = (I_{8 n^{2} - 4 n} - U_{1}^{†} U_{1}) U_{2}^{T}, \\ K = {(I_{d} + (I_{d} - H^{†} H) U_{2} U_{1}^{†} U_{1}^{† T} U_{2}^{T} (I_{d} - H^{†} H))}^{- 1}, \\ J = H^{†} + (I_{d} - H^{†} H) K U_{2} U_{1}^{†} U_{1}^{† T} (I_{8 n^{2} - 4 n} - U_{2}^{T} H^{†}), \\ R_{11} = I_{d} - U_{1} U_{1}^{†} + U_{1}^{† T} U_{2}^{T} K (I_{d} - H^{†} H) U_{2} U_{1}^{†}, \\ R_{12} = - U_{1}^{† T} U_{2}^{T} (I_{d} - H^{†} H) K, \\ R_{22} = (I_{d} - H^{†} H) K . \end{matrix}

From the findings [27] presented above, it can be inferred that

{(\begin{matrix} U_{1} \\ U_{2} \end{matrix})}^{†} = (U_{1}^{†} - J^{T} U_{2} U_{1}^{†}, J^{T}), {(\begin{matrix} U_{1} \\ U_{2} \end{matrix})}^{†} (\begin{matrix} U_{1} \\ U_{2} \end{matrix}) = U_{1}^{†} U_{1} + H H^{†},

(15)

and

I_{2 d} - (\begin{matrix} U_{1} \\ U_{2} \end{matrix}) {(\begin{matrix} U_{1} \\ U_{2} \end{matrix})}^{†} = (\begin{matrix} R_{11} & R_{12} \\ R_{12}^{T} & R_{22} \end{matrix}) .

(16)

Taking into account the aforementioned results, we then turn our attention to the Hermitian solution of the system (1).

Theorem 5.

Let

A_{1}, A_{2}, A_{3}, C_{1}, C_{2}, C_{3} \in H_{c}^{m \times n}, B_{1}, B_{2}, B_{3}, D_{1}, D_{2}, D_{3} \in H_{c}^{n \times k}, A_{4} \in H_{c}^{s \times n}, B_{4} \in H_{c}^{n \times t}, C_{4} \in H_{c}^{s \times t}, A_{5}, A_{6}, A_{7}, B_{5}, B_{6}, B_{7} \in H_{c}^{n \times n}

and

C_{5} \in H_{c}^{n \times n}

U_{1}, U_{2}

and ϵ are in the form of (Section 3) and (14), respectively. Then the system of commutative quaternion matrix equations (1) has a solution

X, Y, Z, W \in {HH}_{c}^{n \times n}

if and only if

(\begin{matrix} U_{1} \\ U_{2} \end{matrix}) {(\begin{matrix} U_{1} \\ U_{2} \end{matrix})}^{†} ϵ = ϵ .

(17)

In this case, the set of Hermitian solutions is as follows:

Λ = \{(X, Y, Z, W) | (\begin{matrix} vec (\overset{´}{X}) \\ vec (\overset{´}{Y}) \\ vec (\overset{´}{Z}) \\ vec (\overset{´}{W}) \end{matrix}) = W {(\begin{matrix} U_{1} \\ U_{2} \end{matrix})}^{†} ϵ + W (I_{8 n^{2} - 4 n} - {(\begin{matrix} U_{1} \\ U_{2} \end{matrix})}^{†} (\begin{matrix} U_{1} \\ U_{2} \end{matrix})) y\},

(18)

where y is an arbitrary vector of appropriate order. Then the system (1) has a unique solution

(X, Y, Z, W) \in Λ

if and only if

r (\begin{matrix} U_{1} \\ U_{2} \end{matrix}) = 8 n^{2} - 4 n .

(19)

If this condition satisfies, then

Λ = \{(X, Y, Z, W) | (\begin{matrix} vec (\overset{´}{X}) \\ vec (\overset{´}{Y}) \\ vec (\overset{´}{Z}) \\ vec (\overset{´}{W}) \end{matrix}) = W {(\begin{matrix} U_{1} \\ U_{2} \end{matrix})}^{†} ϵ\} .

(20)

Proof.

By virtue of Theorem 1 and Theorem 4 , we obtain

\begin{matrix} \{\begin{matrix} A_{1} X = C_{1}, \\ Y B_{1} = D_{1}, \\ A_{2} Z = C_{2}, Z B_{2} = D_{2}, \\ A_{3} W = C_{3}, W B_{3} = D_{3}, A_{4} W B_{4} = C_{4}, \\ A_{5} X + Y B_{5} + A_{6} Z B_{6} + A_{7} W B_{7} = C_{5}, \end{matrix} \\ ⟺ \{\begin{matrix} Φ_{A_{1} X} = Φ_{C_{1}}, \\ Φ_{Y B_{1}} = Φ_{D_{1}}, \\ Φ_{A_{2} Z} = Φ_{C_{2}}, Φ_{Z B_{2}} = Φ_{D_{2}}, \\ Φ_{A_{3} W} = Φ_{C_{3}}, Φ_{W B_{3}} = Φ_{D_{3}}, Φ_{A_{4} W B_{4}} = Φ_{C_{4}}, \\ Φ_{A_{5} X} + Φ_{Y B_{5}} + Φ_{A_{6} Z B_{6}} + Φ_{A_{7} W B_{7}} = Φ_{C_{5}}, \end{matrix} \\ ⟺ E (\begin{matrix} {vec}_{S} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \end{matrix}) + F (\begin{matrix} {vec}_{S} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \end{matrix}) + P (\begin{matrix} {vec}_{S} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \end{matrix}) + Q (\begin{matrix} {vec}_{S} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \end{matrix}) = T, \\ ⟺ (Re E + i Im E) (\begin{matrix} {vec}_{S} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \end{matrix}) + (Re F + i Im F) (\begin{matrix} {vec}_{S} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \end{matrix}) \\ + (Re P + i Im P) (\begin{matrix} {vec}_{S} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \end{matrix}) + (Re Q + i Im Q) (\begin{matrix} {vec}_{S} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \end{matrix}) = T_{1} + i T_{2}, \end{matrix}

\begin{matrix} ⟺ (\begin{matrix} Re E & Re F & Re P & Re Q \\ Im E & Im F & Im P & Im Q \end{matrix}) (\begin{matrix} vec (\vec{X}) \\ vec (\vec{Y}) \\ vec (\vec{Z}) \\ vec (\vec{W}) \end{matrix}) = ϵ, \\ ⟺ (\begin{matrix} U_{1} \\ U_{2} \end{matrix}) (\begin{matrix} vec (\vec{X}) \\ vec (\vec{Y}) \\ vec (\vec{Z}) \\ vec (\vec{W}) \end{matrix}) = ϵ . \end{matrix}

By Lemma 2, we conclude that the system (1) has a Hermitian solution

(X, Y, Z, W) \in Λ

if and only if (17) satisfy, thus we have

(\begin{matrix} vec (\vec{X}) \\ vec (\vec{Y}) \\ vec (\vec{Z}) \\ vec (\vec{W}) \end{matrix}) = {(\begin{matrix} U_{1} \\ U_{2} \end{matrix})}^{†} ϵ + (I_{8 n^{2} - 4 n} - {(\begin{matrix} U_{1} \\ U_{2} \end{matrix})}^{†} (\begin{matrix} U_{1} \\ U_{2} \end{matrix})) y .

On account of

vec (\overset{´}{X}) = (\begin{matrix} vec (error) \\ vec (error) \\ vec (error) \\ vec (error) \end{matrix}) = (\begin{matrix} K_{S} & 0 & 0 & 0 \\ 0 & K_{A} & 0 & 0 \\ 0 & 0 & K_{A} & 0 \\ 0 & 0 & 0 & K_{A} \end{matrix}) (\begin{matrix} {vec}_{S} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \\ {vec}_{A} (error) \end{matrix}) = W vec (\vec{X}),

similarly, we can derive

vec (\overset{´}{Y}) = W vec (\vec{Y}), vec (\overset{´}{Z}) = W vec (\vec{Z})

and

vec (\overset{´}{W}) = W vec (\vec{W})

, then we have

\begin{matrix} (\begin{matrix} vec (\overset{´}{X}) \\ vec (\overset{´}{Y}) \\ vec (\overset{´}{Z}) \\ vec (\overset{´}{W}) \end{matrix}) & = (\begin{matrix} W & 0 & 0 & 0 \\ 0 & W & 0 & 0 \\ 0 & 0 & W & 0 \\ 0 & 0 & 0 & W \end{matrix}) (\begin{matrix} vec (\vec{X}) \\ vec (\vec{Y}) \\ vec (\vec{Z}) \\ vec (\vec{W}) \end{matrix}) \\ = W (\begin{matrix} vec (\vec{X}) \\ vec (\vec{Y}) \\ vec (\vec{Z}) \\ vec (\vec{W}) \end{matrix}) \\ = W {(\begin{matrix} U_{1} \\ U_{2} \end{matrix})}^{†} ϵ + W (I_{8 n^{2} - 4 n} - {(\begin{matrix} U_{1} \\ U_{2} \end{matrix})}^{†} (\begin{matrix} U_{1} \\ U_{2} \end{matrix})) y . \end{matrix}

It means that (18) is true, if (17) holds, the system (1) has a unique solution

(X, Y, Z, W) \in Λ

if and only if

{(\begin{matrix} U_{1} \\ U_{2} \end{matrix})}^{†} (\begin{matrix} U_{1} \\ U_{2} \end{matrix}) = I_{8 n^{2} - 4 n} .

Thus by (19), we can obtain (20). □

Corollary 1.

The system (1) has a solution

X, Y, Z, W \in {HH}_{c}

if and only if

(\begin{matrix} R_{11} & R_{12} \\ R_{12}^{T} & R_{22} \end{matrix}) ϵ = 0 .

(21)

Under this circumstances, the set of Hermitian solution of the system (1) can be represented as follows:

Λ = \{(X, Y, Z, W) | (\begin{matrix} vec (\overset{´}{X}) \\ vec (\overset{´}{Y}) \\ vec (\overset{´}{Z}) \\ vec (\overset{´}{W}) \end{matrix}) = W (U_{1}^{†} - J^{T} U_{2} U_{1}^{†}, J^{T}) ϵ + W (I_{8 n^{2} - 4 n} - U_{1}^{†} U_{1} - H H^{†}) y\},

(22)

in which y is an arbitrary vector of appropriate size. Then the system (1) has an unique solution

(X, Y, Z, W) \in Λ

when (21) and (19) are obeyed. In this case,

Λ = \{(X, Y, Z, W) | (\begin{matrix} vec (\overset{´}{X}) \\ vec (\overset{´}{Y}) \\ vec (\overset{´}{Z}) \\ vec (\overset{´}{W}) \end{matrix}) = W (U_{1}^{†} - J^{T} U_{2} U_{1}^{†}, J^{T}) ϵ\} .

(23)

4. Numerical exemplification

In this section, on the basis of discussions in Section 2 and Section 3, we provide two algorithms for solving the system (1), and present two numerical examples to verify the feasibility of algorithms.

From the previous theoretical analysis, if the system (1) is solvable, then

θ_{1} = ∥(\begin{matrix} U_{1} \\ U_{2} \end{matrix}) {(\begin{matrix} U_{1} \\ U_{2} \end{matrix})}^{†} ϵ - ϵ∥, θ_{2} = ∥(\begin{matrix} R_{11} & R_{12} \\ R_{12}^{T} & R_{22} \end{matrix}) ϵ∥

and

θ_{3} = ∥I_{2 d} - (\begin{matrix} U_{1} \\ U_{2} \end{matrix}) {(\begin{matrix} U_{1} \\ U_{2} \end{matrix})}^{†} - (\begin{matrix} R_{11} & R_{12} \\ R_{12}^{T} & R_{22} \end{matrix})∥

are small enough.

Example 1.

Let

m = 6, n = 4, k = 5, s = 2, t = 3

, and

\begin{matrix} A_{1} = A_{11} + A_{12} j, A_{2} = A_{21} + A_{22} j, A_{3} = A_{31} + A_{32} j, \\ B_{1} = B_{11} + B_{12} j, B_{2} = B_{21} + B_{22} j, B_{3} = B_{31} + B_{32} j, \\ A_{4} = A_{41} + A_{42} j, B_{4} = B_{41} + B_{42} j, A_{5} = A_{51} + A_{52} j, \\ A_{6} = A_{61} + A_{62} j, A_{7} = A_{71} + A_{72} j, B_{5} = B_{51} + B_{52} j, \\ B_{6} = B_{61} + B_{62} j, B_{7} = B_{71} + B_{72} j, \\ \dot{X} = {\dot{X}}_{1} + {\dot{X}}_{2} j, \dot{Y} = {\dot{Y}}_{1} + {\dot{Y}}_{2} j, \\ \dot{Z} = {\dot{Z}}_{1} + {\dot{Z}}_{2} j, \dot{W} = {\dot{W}}_{1} + {\dot{W}}_{2} j, \\ C_{1} = A_{1} \dot{X}, C_{2} = A_{2} \dot{Z}, C_{3} = A_{3} \dot{W}, \\ D_{1} = \dot{Y} B_{1}, D_{2} = \dot{Z} B_{2}, D_{3} = \dot{W} B_{3}, \\ C_{4} = A_{4} \dot{W} B_{4}, C_{5} = A_{5} \dot{X} + \dot{Y} B_{5} + A_{6} \dot{Z} B_{6} + A_{7} \dot{W} B_{7} . \end{matrix}

where

\begin{matrix} A_{11} = (\begin{matrix} I_{4} \\ O_{2 \times 4} \end{matrix}), A_{12} = O_{6 \times 4}, A_{21} = O_{6 \times 4}, A_{22} = (\begin{matrix} I_{4} \\ O_{2 \times 4} \end{matrix}), \\ A_{31} = O_{6 \times 4}, A_{32} = (\begin{matrix} O_{2 \times 4} \\ I_{4} i \end{matrix}), A_{41} = (\begin{matrix} 1 & 0 & i & 0 \\ 0 & 1 & 0 & i \end{matrix}), A_{42} = (\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}), \\ A_{51} = I_{4} i, A_{52} = O_{4 \times 4}, A_{61} = (\begin{matrix} I_{2} & O_{2 \times 2} \\ O_{2 \times 2} & O_{2 \times 2} \end{matrix}), A_{62} = O_{4 \times 4}, \\ A_{71} = (\begin{matrix} O_{2 \times 2} & I_{2} \\ I_{2} & O_{2 \times 2} \end{matrix}), A_{72} = (\begin{matrix} O_{2 \times 2} & - I_{2} i \\ O_{2 \times 2} & O_{2 \times 2} \end{matrix}), B_{11} = O_{4 \times 5}, B_{12} = (\begin{matrix} I_{4} & O_{4 \times 1} \end{matrix}), \\ B_{21} = (\begin{matrix} - I_{4} i & O_{4 \times 1} \end{matrix}), B_{22} = O_{4 \times 5}, B_{31} = O_{4 \times 5}, B_{32} = (\begin{matrix} I_{4} & O_{4 \times 1} \end{matrix}), \\ B_{41} = (\begin{matrix} I_{3} \\ O_{1 \times 3} \end{matrix}), B_{42} = O_{4 \times 3}, B_{51} = O_{4 \times 4}, B_{52} = - I_{4} i, \\ B_{61} = I_{4}, B_{62} = O_{4 \times 4}, B_{71} = (\begin{matrix} O_{2 \times 2} & I_{2} \\ I_{2} & O_{2 \times 2} \end{matrix}), B_{72} = O_{4 \times 4}, \\ C_{1} = (\begin{matrix} 1 & - 1 + i & j & 0 \\ - 1 - i & 2 & i & k \\ - j & - i & 0 & 0 \\ 0 & - k & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), C_{2} = (\begin{matrix} j & 0 & 0 & k \\ 0 & j & 0 & - 1 \\ 0 & 0 & j & 0 \\ - k & 1 & 0 & j \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), \end{matrix}

\begin{matrix} C_{3} = (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & - j & i & 0 \\ j & 0 & k & 0 \\ - i & k & 0 & - 1 \\ 0 & 0 & 1 & 0 \end{matrix}), D_{1} = (\begin{matrix} 0 & 0 & - k & j & 0 \end{matrix}), \\ D_{2} = (\begin{matrix} - i & 0 & 0 & 1 & 0 \\ 0 & - i & 0 & k & 0 \\ 0 & 0 & - i & 0 & 0 \\ 0 & k & j & 0 & 0 \\ - k & 0 & - 1 + k & 0 & 0 \\ j & 1 - k & 0 & k & 0 \\ - 1 & - k & 0 & - i & 0 \end{matrix}), D_{3} = (\begin{matrix} 0 & k & 1 & 0 & 0 \\ - k & 0 & j & 0 & 0 \\ - 1 & j & 0 & i & 0 \\ 0 & 0 & - i & 0 & 0 \end{matrix}), \\ C_{4} = (\begin{matrix} - 1 - k & 2 i + j & j \\ - i & 0 & 1 - i + j \end{matrix}), \\ C_{5} = (\begin{matrix} 1 + i & - i + j + k & i - j & 1 + i - k \\ - i - j - k & 1 + 2 i & - 1 + i + j & - 2 j \\ j - 2 k & 1 - i - j & 0 & i + j \\ 1 & j & - i - j & i - k \end{matrix}) . \end{matrix}

We take

\begin{matrix} {\dot{X}}_{1} = (\begin{matrix} 1 & - 1 + i & 0 & 0 \\ - 1 - i & 2 & i & 0 \\ 0 & - i & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}), {\dot{X}}_{2} = (\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & i \\ - 1 & 0 & 0 & 0 \\ 0 & - i & 0 & 0 \end{matrix}), \\ {\dot{Y}}_{1} = (\begin{matrix} 0 & i & 1 & 0 \\ - i & 0 & i & 0 \\ 1 & - i & 0 & i \\ 0 & 0 & - i & 1 \end{matrix}), {\dot{Y}}_{2} = (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), \\ {\dot{Z}}_{1} = (\begin{matrix} 1 & 0 & 0 & i \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ - i & 0 & 0 & 1 \end{matrix}), {\dot{Z}}_{2} = (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}), \\ {\dot{W}}_{1} = (\begin{matrix} 0 & i & 0 & 0 \\ - i & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), {\dot{W}}_{2} = (\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ - 1 & 0 & 0 & i \\ 0 & 0 & - i & 0 \end{matrix}) . \end{matrix}

Let

\begin{matrix} Φ_{C_{1}} = Φ_{A_{1}} G (\dot{X}), \\ Φ_{D_{1}} = Φ_{\dot{Y}} G (B_{1}), \\ Φ_{C_{2}} = Φ_{A_{2}} G (\dot{Z}), Φ_{D_{2}} = Φ_{\dot{Z}} G (B_{2}), \\ Φ_{C_{3}} = Φ_{A_{3}} G (\dot{W}), Φ_{D_{3}} = Φ_{\dot{W}} G (B_{3}), Φ_{C_{4}} = Φ_{A_{4}} G (\dot{W}) G (B_{4}), \\ Φ_{C_{5}} = Φ_{A_{5}} G (\dot{X}) + Φ_{\dot{Y}} G (B_{5}) + Φ_{A_{6}} G (\dot{Z}) G (B_{6}) + Φ_{A_{7}} G (\dot{W}) G (B_{7}) . \end{matrix}

From MATLAB and Algorithm 2, we can obtain

r (\begin{matrix} U_{1} \\ U_{2} \end{matrix}) = 112 = 8 n^{2} - 4 n, θ_{2} = 7.0360 \times 10^{- 15} .

According to Algorithm 2, the system of matrix equations (1) has an unique solution

(X, Y, Z, W) \in Λ

, and we derive

θ_{1} = 1.7493 \times 10^{- 14}, θ_{3} = 2.3111 \times 10^{- 14}

and

∥ Φ_{(X, Y, Z, W)} - Φ_{(\dot{X}, \dot{Y}, \dot{Z}, \dot{W})} ∥ = 1.1628 \times 10^{- 14}

Example 2.

Let

m = 2, n = 2, k = 2, s = 2, t = 2

, and

\begin{matrix} A_{1} = (\begin{matrix} i & 0 \\ 0 & 0 \end{matrix}), A_{2} = (\begin{matrix} 1 & i \\ 0 & 0 \end{matrix}), A_{3} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}), A_{4} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}), \\ A_{5} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}), A_{6} = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), A_{7} = (\begin{matrix} 1 & 0 \\ 0 & i \end{matrix}), B_{1} = (\begin{matrix} i & 0 \\ 0 & 0 \end{matrix}), \\ B_{2} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}), B_{3} = (\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}), B_{4} = (\begin{matrix} 0 & k \\ 0 & 0 \end{matrix}), B_{5} = (\begin{matrix} 0 & j \\ 0 & 0 \end{matrix}), \\ B_{6} = O_{2}, B_{7} = (\begin{matrix} 0 & 0 \\ 0 & j \end{matrix}), C_{1} = (\begin{matrix} i & - 1 + i \\ 0 & 0 \end{matrix}), C_{2} = (\begin{matrix} 0 & i \\ 0 & 0 \end{matrix}), \\ C_{3} = (\begin{matrix} 0 & 2 + i \\ 2 - i & 0 \end{matrix}), C_{4} = (\begin{matrix} 0 & 0 \\ 0 & j + 2 k \end{matrix}), C_{5} = (\begin{matrix} 1 & 1 + i + 3 j + k \\ 1 - i & - k \end{matrix}), \\ D_{1} = (\begin{matrix} i & 0 \\ 1 & 0 \end{matrix}), D_{2} = (\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}), D_{3} = (\begin{matrix} 0 & 0 \\ 2 - i & 0 \end{matrix}), \end{matrix}

taking

\begin{matrix} X = (\begin{matrix} 1 & 1 + i \\ 1 - i & 0 \end{matrix}), Y = (\begin{matrix} 1 & i \\ - i & 1 \end{matrix}), Z = (\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}), W = (\begin{matrix} 0 & 2 + i \\ 2 - i & 0 \end{matrix}) . \end{matrix}

From MATLAB and Algorithm 2, we obtain

r (\begin{matrix} U_{1} \\ U_{2} \end{matrix}) = 23, θ_{2} = 2.1412 \times 10^{- 15} .

According to Algorithm 2, the system (1) has infinite solutions

(X, Y, Z, W) \in Λ

. We can also obtain

θ_{1} = 3.3532 \times 10^{- 15}, θ_{3} = 4.4730 \times 10^{- 15}

. Then the optimization problem

min_{(X, Y, Z, W) \in Λ} (∥Φ_{(X, Y, Z, W)}∥)

has an unique minimizer

(\tilde{X}, \tilde{Y}, \tilde{Z}, \tilde{W})

, it can also be expressed as

(\begin{matrix} vec (\overset{´}{\tilde{X}}) \\ vec (\overset{´}{\tilde{Y}}) \\ vec (\overset{´}{\tilde{Z}}) \\ vec (\overset{´}{\tilde{W}}) \end{matrix}) = W (U_{1}^{†} - J^{T} U_{2} U_{1}^{†}, J^{T}) ϵ .

Therefore, we can get

∥ Φ_{(X, Y, Z, W)} - Φ_{(\tilde{X}, \tilde{Y}, \tilde{Z}, \tilde{W})} ∥ = 1

, and

\begin{matrix} \tilde{X} = (\begin{matrix} 1 & 1 + i \\ 1 - i & 0 \end{matrix}), \tilde{Y} = (\begin{matrix} 1 & i \\ - i & 0 \end{matrix}), \tilde{Z} = (\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}), \tilde{W} = (\begin{matrix} 0 & 2 + i \\ 2 - i & 0 \end{matrix}) . \end{matrix}

5. Conclusion

In this paper, we have given the necessary and sufficient conditions for the existence of the Hermitian solutions to the system of commutative quaternion matrix equations (1), and also have established an expression of the Hermitian solutions to the system (1) when it is consistent. Some numerical algorithms and examples are provided to illustrate our results.

Author Contributions

Conceptualization, methodology, validation, writing—original draft—Y.Z.; funding acquisition—Q.-W. W.; discussion, writing-final draft—Q.-W.W., L.-M.X.,Y.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported by grants from the National Natural Science Foundation of China (No. 12371023).

Institutional Review Board Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

References

Hamilton, W.R. Lectures on quaternions; Hodges and Smith: Dublin, 1853. [Google Scholar]
Took, C.C.; Mandic, D.P. Augmented second-order statistics of quaternion random signals. Signal Process. 2011, 91, 214–224. [Google Scholar] [CrossRef]
Miao, J.; Kou, K.I. Color image recovery using low-rank quaternion matrix completion algorithm. IEEE Trans. Signal Process. 2021, 31, 190–201. [Google Scholar] [CrossRef]
Jia, Z.G.; Ling, S.T.; Zhao, M.X. Color two-dimensional principal component analysis for face recognition based on quaternion model. Intelligent Computing Theories and Application: 13th International Conference, ICIC 2017, Liverpool, UK, 7-10 August, 2017; Volume 10361, pp. 177–189. [Google Scholar]
Segre, C. The real representations of complex elements and extension to bicomplex systems. Math Ann 1892, 40, 413–467. [Google Scholar] [CrossRef]
Catoni, F. Commutative (segre’s) quaternion fields and relation with maxwell equations. Adv Appl Clifford Al 2008, 18, 9–28. [Google Scholar] [CrossRef]
Pei, S.; Chang, J.H.; Ding, J.J. Commutative reduced biquaternions and their fourier transform for signal and image processing applications. IEEE Trans. Signal Process. 2004, 52, 2012–2031. [Google Scholar] [CrossRef]
Isokawa, T.; Nishimura, H.; Matsui, N. Commutative quaternion and multistate hopfield neural networks. The 2010 International Joint Conference on Neural Networks (IJCNN), Barcelona, Spain, 14 October 2010; Volumn 2000, pp. 1–6. [Google Scholar]
Kobayashi, M. Quaternionic hopfield neural networks with twin-multistate activation function. Neurocomputing 2017, 267, 304–310. [Google Scholar] [CrossRef]
Guo, L.; Zhu, M.; Ge, X. Reduced biquaternion canonical transform, convolution and correlation. Signal Process 2011, 91, 2147–2153. [Google Scholar] [CrossRef]
Kösal, H.; Tosun, M. Commutative quaternion matrices. Adv Appl Clifford Al 2014, 24, 769–779. [Google Scholar] [CrossRef]
Kösal, H.; Akyiğit, M.; Tosun, M. Consimilarity of commutative quaternion matrices. Miskolc Math. Notes 2015, 16, 965–977. [Google Scholar] [CrossRef]
Kösal, H.; Tosun, M. Universal similarity factorization equalities for commutative quaternions and their matrices. Linear Multilinear Algebra 2019, 67, 926–938. [Google Scholar] [CrossRef]
Yu, C.E.; Liu, X.; Zhang, Y. The generalized quaternion matrix equation axb+cx^*d=e. Math. Methods Appl. Sci. 2020, 43, 8506–8517. [Google Scholar] [CrossRef]
Yuan, S.F.; Wang, Q.W.; Yu, Y.B.; Tian, Y. On hermitian solutions of the split quaternion matrix equation axb+cxd=e. Adv Appl Clifford Al 2017, 27, 3235–3252. [Google Scholar] [CrossRef]
Shahzad, A.; Jones, B.; Kerrigan, E.; Constantinides, G. An efficient algorithm for the solution of a coupled sylvester equation appearing in descriptor systems. Automatica 2011, 47, 244–248. [Google Scholar] [CrossRef]
Li, R.C. A bound on the solution to a structured sylvester equation with an application to relative perturbation theory. SIAM J. Matrix Anal. Appl. 1999, 21, 440–445. [Google Scholar] [CrossRef]
Barraud, A.; Lesecq, S.; Christov, N. From sensitivity analysis to random floating point arithmetics - application to sylvester equations. In International Conference on Numerical Analysis and Its Applications; Springer: Berlin, Heidelberg, 2000; Volumn 1988, pp. 35–41. [Google Scholar]
Castelan, E.; Silva, V.G. On the solution of a sylvester equation appearing in descriptor systems control theory. Syst Control Lett 2005, 54, 109–117. [Google Scholar] [CrossRef]
He, Z.H.; Wang, Q.W. A system of periodic discrete-time coupled sylvester quaternion matrix equations. Algebra Colloq 2017, 24, 169–180. [Google Scholar] [CrossRef]
He, Z.H.; Wang, Q.W.; Zhang, Y. A system of quaternary coupled sylvester-type real quaternion matrix equations. Automatica 2018, 87, 25–31. [Google Scholar] [CrossRef]
Wang, Q.W.; He, Z.H.; Zhang, Y. Constrained two-sided coupled sylvester-type quaternion matrix equations. Automatica 2019, 101, 207–213. [Google Scholar] [CrossRef]
Kyrchei, I. Cramer’s rules for Sylvester quaternion matrix equation and its special cases. Adv. Appl. Clifford Algebras 2018, 28, 1–26. [Google Scholar] [CrossRef]
Kyrchei, I. Cramer’s Rules of ηη-(skew-) Hermitian solutions to the quaternion Sylvester-type matrix equations. Adv. Appl. Clifford Algebras 2019, 29, 1–31. [Google Scholar] [CrossRef]
Yuan, S.F.; Wang, Q.W. L-structured quaternion matrices and quaternion linear matrix equations. Linear Multilinear Algebra 2016, 64, 321–339. [Google Scholar] [CrossRef]
Ben-Israel, A.; Greville, T.N. Generalized inverses: theory and applications, 2nd ed.Springer Science & Business Media: New York, 2003. [Google Scholar]
Magnus, J.R. L-structured matrices and linear matrix equations. Linear Multilinear Algebra 1983, 14, 67–88. [Google Scholar] [CrossRef]
Xie, L.M.; Wang, Q.W. A system of matrix equations over the commutative quaternion ring. Filomat 2023, 37, 97–106. [Google Scholar] [CrossRef]
Yuan, S.F.; Tian, Y.; Li, M.Z. On Hermitian solutions of the reduced biquaternion matrix equation (AXB,CXD)=(E,G). Linear Multilinear Algebra 2020, 68, 1355–1373. [Google Scholar] [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

MDPI Initiatives

Important Links

Choose an area of interest and we will send you notifications of new preprints at your preferred frequency.

Disclaimer

The Hermitian Solution to a New System of Commutative Quaternion Matrix Equations

Abstract

1. Introduction

2. Preliminaries

3. The Hermitian solution to the system

4. Numerical exemplification

5. Conclusion

Author Contributions

Funding

Institutional Review Board Statement

Data Availability Statement

Conflicts of Interest

References

MDPI Initiatives

Important Links

Subscribe