1. Introduction
A real quaternion (also called Hamilton quaternion) is usually expressed as
It is a four-dimensional division algebra over the real number field
. During the past decades, matrices over quaternions have played important roles in signal processing, aerospace, color image processing, and many other areas [
1,
2,
3,
4,
5,
6].
Recall that an involutory matrix
R is a square matrix satisfying
, an identity matrix. Clearly,
are trivial involutory matrices. A matrix
is said to be an
-symmetric (resp.
-skew symmetric) matrix if there exist nontrivial involution matrices
R and
S such that
(resp.
) (see [
7,
8]).
-(skew) symmetric matrices are widely used in linear system theory, numerical analysis, and physics [
9,
10,
11,
12,
13,
14]. Trench [
8] discussed the
-(skew) symmetric solutions to the complex matrix equation
. Dehghan and Hajarian [
7] also considered the
-(skew) symmetric solutions to some complex matrix equations. They derived the solvability conditions for the existence of solutions and obtained the general solutions when the matrix equations are solvable. Zhang and Wang [
15] derived the (
)-(skew) symmetric extreme rank solutions to the quaternion system
. Some special cases of (
)-(skew) symmetric matrices, such as (anti-)centrosymmetric matrices,
P-(skew) symmetric matrices, generalized reflexive matrix, and reflexive (antireflexive) matrices, have been discussed [
13,
14,
16,
17,
18]. For instance, Zhou et al. [
18] studied the problems of centrosymmetric matrices over
, where
are the
J, which are on the secondary diagonal and zeros elsewhere. Wang et al. [
17] considered the
P-(skew) symmetric solutions to a pair of quaternion matrix equations.
The matrix equation
is of substantial research significance due to its wide range of applications and its relevance in solving fundamental problems in various disciplines. The (
)-(skew) symmetric matrices include many kinds of important special matrices, like centrosymmetric matrices, P-(skew) symmetric, generalized reflexive matrix, and reflexive (antireflexive) matrices; these kinds of solutions have far-reaching implications in areas ranging from mathematics and engineering to computer science and data analysis [
13,
14,
16,
17,
18].
Motivated by the work and research significance mentioned above, we consider the (
)-(skew) symmetric solutions and least-squares (
)-(skew) symmetric solutions to the quaternion matrix equation
where
are known matrices and
is an unknown matrix.
The paper is organized as follows: In
Section 2, we first introduce some preliminary results. Then we derive the solvability conditions of the equation
for
-(skew) symmetric solutions. In
Section 3, the least-squares
-(skew) symmetric solutions are given. Finally, we provide a numerical example in
Section 4.
Throughout this paper, we propose some notations. For a matrix A, , , and denote the transpose, conjugate transpose, and rank of A separately. Moreover, stands for the Moore–Penrose inverse of A. will be the identity matrix.
2. The Solvability
The real representation method is one of the standard and efficient ways to solve questions over quaternions. There are several real representations; see, for example, [
19,
20,
21]. In this paper, we will use the following
:
For
,
It is easy to verify that this real representation can convert an (
)-(skew) symmetric matrix into a real (
)-(skew) symmetric matrix. For further discussions of our problem, we introduce the following orthogonal matrices:
Now, we summarize some properties of the above real representation in the following lemma:
Lemma 1. Let . Then
- (a)
- (b)
;
- (c)
;
- (d)
- (e)
Let . Then commutes with .
Definition 1. Assume that and are nontrivial involutory matrices. is an -symmetric (resp. -skew symmetric) matrix when A satisfies (resp. ).
Now, we are in a position to derive some necessary and sufficient conditions for the matrix Equation (
1) to have (
)-symmetric (resp.
-skew symmetric) solutions and provide the solutions when the matrix equation is consistent. Let
and
be nontrivial involutory matrices. By (d) of Lemma 1,
thus,
are also nontrivial involutory matrices. For
and
, according to [
8], we can find positive numbers
and matrices
such that
and
For
and
of (
1), we perform the following decomposition:
Now, we have our main results as follows:
Theorem 1. Let , , . Then, there are three equivalent statements:
- (a)
The matrix Equation (1) has an ()-symmetric solution ; - (b)
The matrix equationhas an ()-symmetric solution ; - (c)
The following rank equalities hold:Furthermore, if the matrix Equation (1) is solvable, then
is an ()-symmetric solution to (1), wherewith are arbitrary, and are particular solutions to the matrix equation
. Proof. Let
and
. For the matrix Equation (
1), we convert it into the matrix equation
by the real representation method. First, it can be shown that each (
)-symmetric solution
Y of the real matrix Equation (
3) can generate an (
)-symmetric solution
X of the original matrix Equation (
1).
Suppose (
3) has an (
)-symmetric solution
Y, i.e.,
.
Y may not have the structure of a real representation; thus, we need to construct
with the structure of a real representation from
Y. According to Lemma 1, we have the following three equations:
Since
are nonsingular,
and by Lemma 1(e), we have
Therefore,
,
, and
are also the (
)-symmetric solutions of (
3), and so is
Suppose that
Y can be written in the form of a block matrix:
and substitute it in (
6). Then by computation, we have
where
Now, we construct a new quaternion (
)-symmetric matrix
X using
,
,
, and
:
It is easy to verify that
. By (b) of Lemma 1, we obtain
Thus,
X satisfies (
1). Moreover,
is (
)-symmetric, and we have
It implies that
, and so
X is an (
)-symmetric solution to (
1). Therefore, the consistency of (
3) implies the consistency of (
1). Moreover, any solution
Y of (
3) can generate an (
)-symmetric solution
X of (
1).
Next, we show that if (
1) is consistent, then (
3) is also consistent. Assume (
1) has an (
)-symmetric solution
, i.e.,
. By (b) of Lemma 1, we have
and
Then we can see that
is the (
)-symmetric solution of (
3). Therefore, the original matrix Equation (
1) is consistent if and only if its corresponding real matrix Equation (
3) is consistent. According to Theorem 2.3 in [
7], the real matrix Equation (
3) has an (
)-symmetric solution if and only if the rank equalities in (c) of Theorem 1 hold. Additionally, when (
3) is solvable,
Y in the form of (
5) is the general solution of the real matrix Equation (
3), so we can generate our solution
X by Equation (
4). □
Similar to the proof of Theorem 1, we can obtain the following result for a skew symmetric case.
Theorem 2. Let , , . Then there are three equivalent statements:
- (a)
The matrix Equation (1) has an ()-skew symmetric solution ; - (b)
The matrix equationhas an ()-skew symmetric solution ; - (c)
The following rank equalities hold:
Furthermore, if the matrix Equation (1) is solvable, thenis an ()-skew symmetric solution to (1), wherewith are arbitrary, and are particular solutions to the matrix equation
3. Least-Squares ()-(Skew) Symmetric Solutions
In this section, we derive the least-squares (
)-(skew) symmetric solutions to the quaternion matrix Equation (
1). Let
, where
. We define
We can check that for any
,
To simplify the least-squares (
)-(skew) symmetric solution problems, we are going to use the Frobenius norm of
. By direct calculation, we first derive the relation between
and
:
Let
,
,
. We will find an (
)-(skew) symmetric
such that
The lemma about the least-squares solutions is given as follows:
Lemma 2. [
22]
The solutions of the least-squares problem of the complex matrix equation arein which Z is arbitrary. Now we recall two important decompositions for an ()-symmetric (reps. ()-skew symmetric) matrix over quaternions.
Lemma 3. [
11]
Let be nontrivial involutory matrices with the decompositionThen- (a)
is ()-symmetric if and only if X can be expressed aswhere - (b)
is ()-skew symmetric if and only if Y can be expressed aswhere .
Next, we provide the main result of this section.
Theorem 3. Let , , . Then each least-squares ()-symmetric solution to the matrix equation should be in the form ofwhere E and F are given in Lemma 3, |
|
y = P†b + (I − P†P)z, P = [B1T ⊗ A1, B2T ⊗ A2]τ, b = (Vec(C))cτ, |
01 is the zero matrix with the size of r1r2(m − r1)(n − r2), |
02 is the zero matrix with the size of (m − r1)(n − r2) r1r2, |
I1 is the identity matrix with the size of r1r2 × r1r2, |
I2 is the identity matrix with the size of k × k, |
where k = (m − r1)(n − r2) × (m − r1)(n − r2), |
z ∈
is arbitrary. |
Proof. Using the decomposition of an (
)-symmetric matrix
X in (a) in Lemma 3, we partition the matrices:
where
,
,
,
. Then
has a least-squares (
)-symmetric solution X if and only if
; that is,
has a least-squares solution
. Using the vec operation, we have
which can be rewritten as
Taking the real representations on both sides of (
9),
By Equations (
7) and (
8),
Now, we derive the least-squares solution to the linear system
where
and Lemma 2 tells us that
is the least-squares solution. If we denote
and
then
By the definition of
, we have
Clearly,
is the zero matrix with the size of
, and
is the zero matrix with the size of
. Then
is our required solutions. □
By using the decomposition in (b) of Lemma 3, we can obtain the general expression of the least-squares ()-skew symmetric solution as follows.
Theorem 4. Let , , . Then each least-squares ()-skew symmetric solution to the matrix equation should be in the form ofwhere E and F are given in Lemma 3, and and are block matrices determined by and , respectively. |
|
y = P†b + (I − P†P)z, P = [B2T ⊗ A1, B1T ⊗ A2]τ, b = (Vec(C))cτ, |
03 is the zero matrix with the size of r1(n − r2) × (m − r1)r2, |
04 is the zero matrix with the size of (m − r1)r2 × r1(n − r2), |
I3 is the identity matrix with the size of r1(n − r2) × r1(n − r2), |
I4 is the identity matrix with the size of (m − r1)r2 × (m − r1)r2, |
z ∈
is arbitrary. |