# The Generalized Bisymmetric (Bi-Skew-Symmetric) Solutions of a Class of Matrix Equations and Its Least Squares Problem

Authors: Yifen Ke; Changfeng Ma
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The solvability conditions and the general expression of the generalized bisymmetric and bi-skew-symmetric solutions of a class of matrix equations(
A
X
=
B, X
C
=
D
) are established, respectively. If the solvability conditions are not satisfied, the generalized bisymmetric and bi-skew-symmetric least squares solutions of the matrix equations are considered. In addition, two algorithms are provided to compute the generalized bisymmetric and bi-skew-symmetric least squares solutions. Numerical experiments illustrate that the results are reasonable.


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## Body

## 1.  Introduction

The class of matrix equations, namely,A
X
=
B, X
C
=
D, where A, B, C, and D are given, is one of the most interesting and intensively studied classes of linear algebra. It has been investigated by many authors and a series of important and useful results has been obtained (see, e.g., [1–31]).For example, Cecioni [1] gave a necessary and sufficient condition for the matrix equations to have a common solution and a general expression of the common solution was obtained by Rao and Mitra [2, page 25]; Mitra [3] obtained a common solution with the minimum possible rank and also other feasible specified rank; Chu [4] achieved new necessary and sufficient conditions for the matrix equations by using the generalized singular value decomposition. In [5], Wang and Yu derived the necessary and sufficient conditions and the expressions for the orthogonal solutions, the symmetric orthogonal solutions, and the skew-symmetric orthogonal solutions of the matrix equations, respectively. Khatri and Mitra [6] considered the general Hermitian and nonnegative definite solutions of the matrix equations, respectively. Dajić and Koliha [7] studied the positive solutions to the matrix equations for Hilbert space operators using generalized inverses, and a sufficient and necessary condition for its solvability and a representation of its general solutions were also established therein. Li et al. investigated the generalized reflexive and antireflexive solution of the matrix equations, in [8, 9], respectively. In [10], Qiu et al. considered the unknown matrix X with the constraint P
X
=
s
X
P, where P is a given Hermitian matrix satisfying P
2
=
I and s
=
±
1.In this paper,R
m
×
n, O
R
n
×
n, S
R
n
×
n,  and  S
O
R
n
×
n denote the set of all m
×
n real matrices, the set of all n
×
n real orthogonal matrices, the set of all n
×
n real symmetric matrices, and the set of all n
×
n real symmetric orthogonal matrices, respectively. A
⊤ represents the transpose of the real matrix A and ∥
·
∥ stands for the Frobenius norm induced by the inner product. (
A
B
) denotes a row block matrix and A
∘
B denotes the Hadamard product produced by A and B, namely, A
∘
B
=
(
a
i
j
b
i
j
). The symbol I
n stands for the identity matrix of order n. Let A
† be the Moore-Penrose generalized inverse of a matrix A
∈
R
m
×
n, which is defined to be the unique solution X
∈
R
n
×
m satisfying the following four matrix equations:
(1)
(
1
)
A
X
A
=
A
,
(
2
)
X
A
X
=
X
,
(
3
)
(
A
X
)
⊤
=
A
X
,
(
4
)
(
X
A
)
⊤
=
X
A
.
Furthermore, ℒ
A and ℛ
A represent the two orthogonal projectors ℒ
A
=
I
n
-
A
†
A and ℛ
A
=
I
m
-
A
A
†. Set S
n
=
(
e
n
,
e
n
-
1
,
…
,
e
1
), where e
i denotes the ith column of the identity matrix I
n. It is easy to see that S
n
⊤
=
S
n and S
n
⊤
S
n
=
I
n.Definition 1.
LetP
∈
S
O
R
n
×
n; that is, P
=
P
⊤
=
P
-
1. A matrix X
∈
R
n
×
n is said to be a generalized bisymmetric matrix if X
=
X
⊤
=
P
X
P. The set of all n
×
n generalized bisymmetric matrices is denoted by G
B
S
R
n
×
n.Definition 2.
LetP
∈
S
O
R
n
×
n; that is, P
=
P
⊤
=
P
-
1. A matrix X
∈
R
n
×
n is said to be a generalized bi-skew-symmetric matrix if X
=
-
X
⊤
=
-
P
X
P. The set of all n
×
n generalized bi-skew-symmetric matrices is denoted by G
B
S
S
R
n
×
n.Without special statement, we assume thatP is a given symmetric orthogonal matrix in the remainder of this paper.IfP
=
S
n, then the generalized bisymmetric matrix reduces to the bisymmetric matrix and the generalized bi-skew-symmetric matrix reduces to the antisymmetric and persymmetric matrix.In this paper, we consider the following problems.Problem 3.
GivenA
,
B
∈
R
m
×
n, C
,
D
∈
R
n
×
l, find X
∈
G
B
S
R
n
×
n such that
(2)
A
X
=
B
,
X
C
=
D
.Problem 4.
GivenA
,
B
∈
R
m
×
n, C
,
D
∈
R
n
×
l, find X
∈
G
B
S
S
R
n
×
n such that
(3)
A
X
=
B
,
X
C
=
D
.Problem 5.
GivenA
,
B
∈
R
m
×
n, C
,
D
∈
R
n
×
l, find X
^
∈
G
B
S
R
n
×
n such that
(4)
∥
A
X
^
-
B
∥
+
∥
X
^
C
-
D
∥
=
min
⁡
X
∈
G
B
S
R
n
×
n
∥
A
X
-
B
∥
+
∥
X
C
-
D
∥
.Problem 6.
GivenA, B
∈
R
m
×
n, C, D
∈
R
n
×
l, find X
^
∈
G
B
S
S
R
n
×
n such that
(5)
∥
A
X
^
-
B
∥
+
∥
X
^
C
-
D
∥
=
min
⁡
X
∈
G
B
S
S
R
n
×
n
∥
A
X
-
B
∥
+
∥
X
C
-
D
∥
.IfC
=
D
=
0, then those problems become the problems discussed in [11]. So, this paper extends the part results of [11]. In our work, the necessary and sufficient conditions for the existence of the solutions to Problems 3 and 4 are derived and their general expressions of the solutions are given by Moore-Penrose generalized inverse, respectively. If the solvability conditions are not satisfied, Problems 5 and 6 will be considered.The remainder of this paper is arranged as follows. In Section2, we establish the necessary and sufficient conditions and the explicit expressions of Problems 3 and 4. In Section 3, we investigate Problems 5 and 6 by virtue of the singular value decomposition (SVD) and the special decompositions of the generalized bisymmetric matrices and the generalized bi-skew-symmetric matrices. In Section 4, we give two algorithms and some examples to illustrate the efficiency of our proposed results. In Section 5, some conclusions are made.


## 2. The Generalized Bisymmetric (Bi-Skew-Symmetric) Solutions of the Matrix EquationsA
X
=
B
,
X
C
=
D

In this section, we first recall some lemmas which will be used for obtaining the necessary and sufficient conditions and the explicit expressions of Problems3 and 4.Lemma 7 (see [12]).
AssumeP
∈
S
O
R
n
×
n, and let(6)
P
1
=
1
2
(
I
n
+
P
)
,
P
2
=
1
2
(
I
n
-
P
)
.
Then P
1 and P
2 are orthogonal projection matrices satisfying P
1
+
P
2
=
I
n, P
1
P
2
=
0. Furthermore, assume rank
⁡
(
P
1
)
=
r. Then, rank
⁡
(
P
2
)
=
n
-
r and there exist unit column orthogonal matrices U
1
∈
R
n
×
r and U
2
∈
R
n
×
(
n
-
r
) such that
(7)
P
1
=
U
1
U
1
⊤
,
P
2
=
U
2
U
2
⊤
,
P
=
U
1
U
1
⊤
-
U
1
U
1
⊤
,
U
1
⊤
U
2
=
0
.From Lemma7, we note that U
=
(
U
1
U
2
) is an orthogonal matrix and the symmetric orthogonal matrix P can be expressed as
(8)
P
=
U
(
I
r
0
0
-
I
n
-
r
)
U
⊤
.Lemma 8 (see [11]).
Assume that the spectral decomposition ofP is given as in (8). Then X
∈
G
B
S
R
n
×
n if and only if X can be expressed as
(9)
X
=
U
(
X
11
0
0
X
22
)
U
⊤
,
where X
11
∈
S
R
r
×
r and X
22
∈
S
R
(
n
-
r
)
×
(
n
-
r
).Proof.
ForX
∈
G
B
S
R
n
×
n, by Definition 1, it is easy to know that P
X
=
X
P. Then, we have
(10)
P
1
X
P
1
=
I
n
+
P
2
X
I
n
+
P
2
=
1
4
(
X
+
P
X
+
X
P
+
P
X
P
)
=
1
2
(
X
+
P
X
)
,
P
2
X
P
2
=
I
n
-
P
2
X
I
n
-
P
2
=
1
4
(
X
-
P
X
-
X
P
+
P
X
P
)
=
1
2
(
X
-
P
X
)
.
By (10) and Lemma 7, we obtain
(11)
X
=
P
1
X
P
1
+
P
2
X
P
2
=
U
1
U
1
⊤
X
U
1
U
1
⊤
+
U
2
U
2
⊤
X
U
2
U
2
⊤
.
Let X
11
=
U
1
⊤
X
U
1 and X
22
=
U
2
⊤
X
U
2; it is easy to verify that X
11
=
X
11
⊤ and X
22
=
X
22
⊤. Furthermore, we have
(12)
X
=
U
1
X
11
U
1
⊤
+
U
2
X
22
U
2
⊤
=
U
(
X
11
0
0
X
22
)
U
⊤
.
Conversely, for anyX
11
∈
S
R
r
×
r and X
22
∈
S
R
(
n
-
r
)
×
(
n
-
r
), it is easy to verify that X
=
X
⊤. Using (8), we have
(13)
P
X
P
=
U
(
I
r
0
0
-
I
n
-
r
)
U
⊤
U
(
X
11
0
0
X
22
)
U
⊤
U
(
I
r
0
0
-
I
n
-
r
)
U
⊤
wiiii
=
U
(
X
11
0
0
X
22
)
U
⊤
wiiii
=
X
.
This implies that
(14)
X
=
U
(
X
11
0
0
X
22
)
U
⊤
∈
G
B
S
R
n
×
n
.Lemma 9 (see [11]).
Assume that the spectral decomposition ofP is given as in (8). Then, X
∈
G
B
S
S
R
n
×
n if and only if X can be expressed as
(15)
X
=
U
(
0
X
12
-
X
12
⊤
0
)
U
⊤
,
where X
12
∈
R
r
×
(
n
-
r
).Proof.
ForX
∈
G
B
S
S
R
n
×
n, by Definition 2, we have
(16)
P
1
X
P
1
=
I
n
+
P
2
X
I
n
+
P
2
=
1
4
(
X
+
P
X
+
X
P
+
P
X
P
)
=
1
4
(
P
X
+
X
P
)
,
P
2
X
P
2
=
I
n
-
P
2
X
I
n
-
P
2
=
1
4
(
X
-
P
X
-
X
P
+
P
X
P
)
=
-
1
4
(
P
X
+
P
X
)
.
By (16) and Lemma 7, we obtain
(17)
X
=
(
P
1
+
P
2
)
X
(
P
1
+
P
2
)
=
P
1
X
P
1
+
P
1
X
P
2
+
P
2
X
P
1
+
P
2
X
P
2
=
P
1
X
P
2
+
P
2
X
P
1
=
U
1
U
1
⊤
X
U
2
U
2
⊤
+
U
2
U
2
⊤
X
U
1
U
1
⊤
.
Let X
12
=
U
1
⊤
X
U
2 and X
21
=
U
2
⊤
X
U
1; it is easy to verify that X
21
=
-
X
12
⊤. And we have
(18)
X
=
U
1
X
12
U
2
⊤
+
U
2
X
21
U
1
⊤
=
U
(
0
X
12
X
21
0
)
U
⊤
=
U
(
0
X
12
-
X
12
⊤
0
)
U
⊤
.
Conversely, for anyX
12
∈
R
r
×
(
n
-
r
), it is easy to verify that X
=
-
X
⊤. Using (8), we have
(19)
P
X
P
=
U
(
I
r
0
0
-
I
n
-
r
)
U
⊤
U
(
0
X
12
-
X
12
⊤
0
)
U
⊤
U
(
I
r
0
0
-
I
n
-
r
)
U
⊤
wiiii
=
U
(
0
-
X
12
X
12
⊤
0
)
U
⊤
wiiii
=
-
X
.
This implies that
(20)
X
=
U
(
0
X
12
-
X
12
⊤
0
)
U
⊤
∈
G
B
S
S
R
n
×
n
.Remark 10.
For Lemma8, Wang and Yu [11] just gave the conclusion; we prove it here. The proof of Lemma 9 can be seen in [12]; for the convenience of the reader, we rewrite it.Lemma 11 (see [13]).
Suppose thatA
1
∈
R
m
×
n, A
3
∈
R
k
×
n, B
2
∈
R
r
×
s, B
4
∈
R
r
×
l, C
1
∈
R
m
×
r, C
2
∈
R
n
×
s, C
3
∈
R
k
×
r, C
4
∈
R
n
×
l are known and X
∈
R
n
×
r is unknown. Let K
=
A
3
ℒ
A
1, N
=
ℛ
B
2
B
4, Q
1
=
C
3
-
A
3
A
1
†
C
1
-
K
C
2
B
2
†, Q
=
C
4
-
A
1
†
C
1
B
4
-
ℒ
A
1
C
2
B
2
†
B
4
-
ℒ
A
1
K
†
Q
1
N. Then the system of matrix equations
(21)
A
1
X
=
C
1
,
X
B
2
=
C
2
,
A
3
X
=
C
3
,
X
B
4
=
C
4
is consistent if and only if
(22)
K
K
†
Q
1
ℛ
B
2
=
Q
1
,
Q
ℒ
N
=
0
,
ℛ
ℒ
A
1
ℒ
K
Q
=
0
,
A
1
C
2
=
C
1
B
2
,
A
i
A
i
†
C
i
=
C
i
,
C
j
B
j
†
B
j
=
C
j
,
i
=
1,3
;
j
=
2,4
,
in which case, the general solutions of the system can be expressed as
(23)
X
=
A
1
†
C
1
+
ℒ
A
1
C
2
B
2
†
+
ℒ
A
1
K
†
Q
1
ℛ
B
2
w
+
Q
N
†
ℛ
B
2
+
ℒ
A
1
ℒ
K
Z
ℛ
N
ℛ
B
2
,
where Z is an arbitrary real matrix with compatible dimension.Lemma 12.
GivenA
,
B
∈
R
m
×
n, C
,
D
∈
R
n
×
l. Let K
=
C
⊤
ℒ
A, N
=
ℛ
C
A
⊤, Q
1
=
D
⊤
-
C
⊤
A
†
B
-
K
D
C
†, Q
=
B
⊤
-
A
†
B
A
⊤
-
ℒ

A
D
C
†
A
⊤
-
ℒ

A
K
†
Q
1
N. Then, the following statements are equivalent.(i) The matrix equations(24)
A
X
=
B
,
X
C
=
D have a solution X
∈
S
R
n
×
n.(ii) The system of matrix equations(25)
A
Y
=
B
,
Y
C
=
D
,
Y
A
⊤
=
B
⊤
,
C
⊤
Y
=
D
⊤ have a solution Y
∈
R
n
×
n; in this case, the symmetric solution of the matrix equations (24) is
(26)
X
=
Y
+
Y
⊤
2
.
(iii) If(27)
K
K
†
Q
1
ℛ
C
=
Q
1
,
Q
ℒ
N
=
0
,
ℛ
ℒ
A
ℒ
K
Q
=
0
A
D
=
B
C
,
A
A
†
B
=
B
,
D
C
†
C
=
D
,the symmetric solutions of the matrix equations (24) can be expressed as
(28)
X
=
1
2
(
A
†
B
+
ℒ
A
D
C
†
+
ℒ
A
K
†
Q
1
ℛ
C
h
h
h
h
+
Q
N
†
ℛ
C
+
ℒ
A
ℒ
K
Z
ℛ
N
ℛ
C
)
i
h
+
1
2
(
B
⊤
A
⊤
†
+
C
⊤
†
D
⊤
ℒ
A
+
ℛ
C
Q
1
⊤
K
⊤
†
ℒ
A
h
h
h
h
h
h
+
ℛ
C
N
⊤
†
Q
⊤
+
ℛ
C
ℛ
N
Z
⊤
ℒ
K
ℒ
A
)
,
where Z
∈
R
n
×
n is an arbitrary matrix.Proof.
(
i
)
⇔
(
ii
). It is not difficult to get that (
i
) is equivalent to (
ii
). Further, if Y is a solution of the matrix equations (25), then(29)
X
=
Y
+
Y
⊤
2
=
X
⊤
.
Moreover,
(30)
A
X
=
A
Y
+
Y
⊤
2
=
1
2
(
A
Y
+
A
Y
⊤
)
=
1
2
(
B
+
B
)
=
B
,
X
C
=
Y
+
Y
⊤
2
C
=
1
2
(
Y
C
+
Y
⊤
C
)
=
1
2
(
D
+
D
)
=
D
.
Then, the expression in (26) is the symmetric solution of the matrix equations (24).
(
ii
)
⇔
(
iii
). From Lemma 11, it can be proved that (
ii
) is equivalent to (
iii
) and the solutions of the matrix equations (25) can be expressed as
(31)
Y
=
A
†
B
+
ℒ
A
D
C
†
+
ℒ
A
K
†
Q
1
ℛ
C
w
+
Q
N
†
ℛ
C
+
ℒ
A
ℒ
K
Z
ℛ
N
ℛ
C
.
Substituting (31) into (26) yields (28). The proof is completed.ForU
∈
O
R
n
×
n which is given by (8), partition
(32)
A
U
=
(
A
1
A
2
)
,
B
U
=
(
B
1
B
2
)
,
h
A
1
,
B
1
∈
R
m
×
r
,
A
2
,
B
2
∈
R
m
×
(
n
-
r
)
,
(33)
C
⊤
U
=
(
C
1
⊤
C
2
⊤
)
,
D
⊤
U
=
(
D
1
⊤
D
2
⊤
)
,
h
h
h
h
h
h
C
1
,
D
1
∈
R
r
×
l
,
C
2
,
D
2
∈
R
(
n
-
r
)
×
l
.
From Lemma 8, we know that the matrix equations A
X
=
B, X
C
=
D have a solution X
∈
G
B
S
R
n
×
n if and only if there exist X
11
∈
S
R
r
×
r and X
22
∈
S
R
(
n
-
r
)
×
(
n
-
r
) such that
(34)
A
U
(
X
11
0
0
X
22
)
=
B
U
,
(
X
11
0
0
X
22
)
U
⊤
C
=
U
⊤
D
,
that is,
(35)
A
1
X
11
=
B
1
,
X
11
C
1
=
D
1
,
(36)
A
2
X
22
=
B
2
,
X
22
C
2
=
D
2
.LetK
1
=
C
1
⊤
ℒ
A
1, N
1
=
ℛ
C
1
A
1
⊤, Q
1
=
D
1
⊤
-
C
1
⊤
A
1
†
B
1
-
K
1
D
1
C
1
†, Q
¯
=
B
1
⊤ − A
1
†
B
1
A
1
⊤ − ℒ
A
1
D
1
C
1
†
A
1
⊤ − ℒ
A
1
K
1
†
Q
1
N
1. By Lemma 12, the matrix equations (35) have a solution X
11
∈
S
R
r
×
r if and only if
(37)
K
1
K
1
†
Q
1
ℛ
C
1
=
Q
1
,
Q
¯
ℒ
N
1
=
0
,
ℛ
ℒ
A
1
ℒ
K
1
Q
¯
=
0
,
A
1
D
1
=
B
1
C
1
,
A
1
A
1
†
B
1
=
B
1
,
D
1
C
1
†
C
1
=
D
1
,
in which case the general solutions can be expressed as
(38)
X
11
=
1
2
(
A
1
†
B
1
+
ℒ
A
1
D
1
C
1
†
+
ℒ

A
1
K
1
†
Q
1
ℛ
C
1
h
h
h
h
+
Q
¯
N
1
†
ℛ
C
1
+
ℒ

A
1
ℒ

K
1
Z
1
ℛ
N
1
ℛ
C
1
)
w
+
1
2
(
Q
¯
⊤
B
1
⊤
A
1
⊤
†
+
C
1
⊤
†
D
1
⊤
ℒ

A
1
+
ℛ
C
1
Q
1
⊤
K
1
⊤
†
ℒ
A
1
h
h
h
h
h
h
+
ℛ
C
1
N
1
⊤
†
Q
¯
⊤
+
ℛ
C
1
ℛ
N
1
Z
1
⊤
ℒ

K
1
ℒ

A
1
)
,
where Z
1
∈
R
r
×
r is an arbitrary matrix.LetK
2
=
C
2
⊤
ℒ
A
2, N
2
=
ℛ
C
2
A
2
⊤, Q
2
=
D
2
⊤
-
C
2
⊤
A
2
†
B
2
-
K
2
D
2
C
2
†, Q
~
=
B
2
⊤ − A
2
†
B
2
A
2
⊤
-
ℒ

A
2
D
2
C
2
†
A
2
⊤
-
ℒ

A
2
K
2
†
Q
2
N
2. Similarly, the matrix equations (36) have a solution X
22
∈
S
R
(
n
-
r
)
×
(
n
-
r
) if and only if
(39)
K
2
K
2
†
Q
2
ℛ
C
2
=
Q
2
,
Q
~
ℒ
N
2
=
0
,
ℛ
ℒ
A
2
ℒ
K
2
Q
~
=
0
,
A
2
D
2
=
B
2
C
2
,
A
2
A
2
†
B
2
=
B
2
,
D
2
C
2
†
C
2
=
D
2
,
in which case the general solutions can be expressed as
(40)
X
22
=
1
2
(
A
2
†
B
2
+
ℒ

A
2
D
2
C
2
†
+
ℒ

A
2
K
2
†
Q
2
ℛ
C
2
h
h
h
h
h
+
Q
~
N
2
†
ℛ
C
2
+
ℒ
A
2
ℒ

K
2
Z
2
ℛ
N
2
ℛ
C
2
)
w
+
1
2
(
B
2
⊤
A
2
⊤
†
+
C
2
⊤
†
D
2
⊤
ℒ

A
2
+
ℛ
C
2
Q
2
⊤
K
2
⊤
†
ℒ
A
2
h
h
h
h
h
h
+
ℛ
C
2
N
2
⊤
†
Q
~
⊤
+
ℛ
C
2
ℛ
N
2
Z
2
⊤
ℒ
K
2
ℒ

A
2
)
,
where Z
2
∈
R
(
n
-
r
)
×
(
n
-
r
) is an arbitrary matrix.Now, based on the above discussion, we give the solvability conditions and the general expression of the solutions of Problem3.Theorem 13.
GivenA
,
B
∈
R
m
×
n, C
,
D
∈
R
n
×
l. And U
∈
O
R
n
×
n is given by (8). Let the partitions of A
U, B
U, C
⊤
U, and D
⊤
U be as in (32) and (33), respectively. Then, Problem 3 is consistent if and only if (37) and (39) hold, in which case the general solutions can be expressed as
(41)
X
=
U
(
X
11
0
0
X
22
)
U
⊤
,
where X
11 and X
22 are given as in (38) and (40).From Lemma9, we know that the matrix equations A
X
=
B, X
C
=
D have a solution X
∈
G
B
S
S
R
n
×
n if and only if there exists X
12
∈
R
r
×
(
n
-
r
) such that
(42)
A
U
(
0
X
12
-
X
12
⊤
0
)
=
B
U
,
(
0
X
12
-
X
12
⊤
0
)
U
⊤
C
=
U
⊤
D
,
that is,
(43)
A
1
X
12
=
B
2
,
X
12
A
2
⊤
=
-
B
1
⊤
,
C
1
⊤
X
12
=
-
D
2
⊤
,
X
12
C
2
=
D
1
.LetK
=
C
1
⊤
ℒ
A
1, N
=
ℛ
A
2
⊤
C
2
=
ℒ
A
2
C
2, Q
1
=
-
D
2
⊤
-
C
1
⊤
A
1
†
B
2
+
K
B
1
⊤
A
2
⊤
†, Q
=
D
1
-
A
1
†
B
2
C
2
+
ℒ
A
1
B
1
⊤
A
2
⊤
†
C
2
-
ℒ
A
1
K
†
Q
1
N. By Lemma 11, the system of matrix equations (43) has a solution X
12
∈
R
r
×
(
n
-
r
) if and only if
(44)
K
K
†
Q
1
ℛ
A
2
⊤
=
Q
1
,
Q
ℒ
N
=
0
,
ℛ
ℒ
A
1
ℒ
K
Q
=
0
,
A
1
B
1
⊤
=
-
B
2
A
2
⊤
,
A
1
A
1
†
B
2
=
B
2
,
D
2
C
1
†
C
1
=
D
2
,
A
2
A
2
†
B
1
=
B
1
,
D
1
C
2
†
C
2
=
D
1
,
in which case the general solutions can be expressed as
(45)
X
12
=
A
1
†
B
2
-
ℒ
A
1
B
1
⊤
A
2
⊤
†
+
ℒ
A
1
K
†
Q
1
ℛ
A
2
⊤
+
Q
N
†
ℛ
A
2
⊤
+
ℒ
A
1
ℒ
K
Z
ℛ
N
ℛ
A
2
⊤
,
where Z
∈
R
n
×
(
n
-
r
) is an arbitrary matrix.Therefore, we have the following result about Problem4.Theorem 14.
GivenA
,
B
∈
R
m
×
n, C
,
D
∈
R
n
×
l. And U
∈
O
R
n
×
n is given by (8). Let the partitions of A
U, B
U, C
⊤
U, and D
⊤
U be as in (32) and (33), respectively. Then, Problem 4 is consistent if and only if (44) holds, in which case the general solutions can be expressed as
(46)
X
=
U
(
0
X
12
-
X
12
⊤
0
)
U
⊤
,
where X
12 is given as in (45).


## 3. The Generalized Bisymmetric (Bi-Skew-Symmetric) Least Squares Solutions of the Matrix EquationsA
X
=
B
,
X
C
=
D

It is well known that if the solvability conditions of the linear matrix equation or linear matrix equations are not satisfied, we can derive its approximate solutions, among which, the least squares solution is usually considered. In this section, we try to solve the Problems5 and 6. Firstly, we present some lemmas which will play important roles in the following.Lemma 15 (see [14]).
GivenA
,
B
∈
R
m
×
n. Let the singular value decomposition (SVD) of A be
(47)
A
=
W
(
Σ
0
0
0
)
V
⊤
,
where W
=
(
W
1
W
2
)
∈
O
R
m
×
m, V
=
(
V
1
V
2
)
∈
O
R
n
×
n, Σ
=
diag
⁡
(
σ
1
,
…
,
σ
s
)
>
0, s
=
rank
⁡
(
A
). Then, there exists X
^
∈
S
R
m
×
m such that
(48)
∥
X
^
A
-
B
∥
=
min
⁡
X
∈
S
R
m
×
m
∥
X
A
-
B
∥
.
In this case, X
^ can be expressed as
(49)
X
^
=
W
(
Φ
∘
(
W
1
⊤
B
V
1
Σ
+
Σ
V
1
⊤
B
⊤
W
1
)
Σ
-
1
V
1
⊤
B
⊤
W
2
W
2
⊤
B
V
1
Σ
-
1
G
)
W
⊤
,
where Φ
=
(
φ
i
j
)
∈
R
s
×
s, φ
i
j
=
1
/
(
σ
i
2
+
σ
j
2
), 1
≤
i, j
≤
s, and G
∈
S
R
(
m
-
s
)
×
(
m
-
s
) is an arbitrary matrix.Lemma 16 (see [15]).
GivenA
∈
R
m
×
n, B
∈
R
m
×
p, C
∈
R
p
×
l, and D
∈
R
n
×
l. Let the SVDs of A and C be
(50)
A
=
W
(
Σ
0
0
0
)
V
⊤
,
C
=
T
(
Ω
0
0
0
)
Q
⊤
,
where W
=
(
W
1
W
2
), V
=
(
V
1
V
2
), T
=
(
T
1
T
2
), and Q
=
(
Q
1
Q
2
) are all orthogonal matrices and the partitions are compatible with the sizes of
(51)
Σ
=
diag
⁡
(
σ
1
,
…
,
σ
s
)
>
0
,
Ω
=
diag
⁡
(
ω
1
,
…
,
ω
t
)
>
0
,
s
=
rank
⁡
(
A
)
,
t
=
rank
⁡
(
C
)
.
Then the least squares solutions of the matrix equations A
X
=
B, X
C
=
D can be expressed as
(52)
X
^
=
V
(
Φ
∘
[
V
1
⊤
(
A
⊤
B
+
D
C
⊤
)
T
1
]
(
Σ
-
1
)
2
V
1
⊤
(
A
⊤
B
+
D
C
⊤
)
T
2
V
2
⊤
(
A
⊤
B
+
D
C
⊤
)
T
1
(
Ω
-
1
)
2
G
)
T
⊤

,
where Φ
=
(
φ
i
j
)
∈
R
s
×
t, φ
i
j
=
1
/
(
σ
i
2
+
ω
j
2
), 1
≤
i
≤
s, 1
≤
j
≤
t, and G
∈
R
(
n
-
s
)
×
(
p
-
t
) is an arbitrary matrix.Let the SVDs of(
A
1
⊤
C
1
) and (
A
2
⊤
C
2
) be
(53)
(
A
1
⊤
C
1
)
=
V
(
Σ
0
0
0
)
W
⊤
,
(
A
2
⊤
C
2
)
=
T
(
Ω
0
0
0
)
Q
⊤
,
where V
=
(
V
1
V
2
), W
=
(
W
1
W
2
), T
=
(
T
1
T
2
), and Q
=
(
Q
1
Q
2
) are all orthogonal matrices and the partitions are compatible with the sizes of
(54)
Σ
=
diag
⁡
(
σ
1
,
…
,
σ
s
)
>
0
,
Ω
=
diag
⁡
(
ω
1
,
…
,
ω
t
)
>
0
,
s
=
rank
⁡
(
(
A
1
⊤
C
1
)
)
,
t
=
rank
⁡
(
(
A
2
⊤
C
2
)
)
.Based on Lemma8 and the properties of Frobenius norm, we have
(55)
∥
A
X
-
B
∥
2
+
∥
X
C
-
D
∥
2
ww
=
∥
A
U
(
X
11
0
0
X
22
)
U
⊤
-
B
∥
2
wwiiii
+
∥
U
(
X
11
0
0
X
22
)
U
⊤
C
-
D
∥
2
ww
=
∥
A
1
X
11
-
B
1
∥
2
+
∥
A
2
X
22
-
B
2
∥
2
wwiiii
+
∥
X
11
C
1
-
D
1
∥
2
+
∥
X
22
C
2
-
D
2
∥
2
ww
=
∥
X
11
(
A
1
⊤
C
1
)
-
(
B
1
⊤
D
1
)
∥
2
wwiiii
+
∥
X
22
(
A
2
⊤
C
2
)
-
(
B
2
⊤
D
2
)
∥
2
.
Hence, min
⁡
X
∈
G
B
S
R
n
×
n
∥
A
X
-
B
∥
+
∥
X
C
-
D
∥ is equivalent to
(56)
min
⁡
X
11
∈
S
R
r
×
r
∥
X
11
(
A
1
⊤
C
1
)
-
(
B
1
⊤
D
1
)
∥
,
(57)
min
⁡
X
22
∈
S
R
(
n
-
r
)
×
(
n
-
r
)
∥
X
22
(
A
2
⊤
C
2
)
-
(
B
2
⊤
D
2
)
∥
.From Lemma15, when X
11 has the form(58)
X
11
=
V
(
Φ
1
∘
[
V
1
⊤
(
B
1
⊤
D
1
)
W
1
Σ
+
Σ
W
1
⊤
(
B
1
⊤
D
1
)
⊤
V
1
]
Σ
-
1
W
1
⊤
(
B
1
⊤
D
1
)
⊤
V
2
V
2
⊤
(
B
1
⊤
D
1
)
W
1
Σ
-
1
G
1
)
V
⊤
,where Φ
1
=
(
φ
i
j
)
∈
R
s
×
s
,
φ
i
j
=
1
/
(
σ
i
2
+
σ
j
2
), 1
≤
i
,
j
≤
s, and G
1
∈
S
R
(
r
-
s
)
×
(
r
-
s
) is an arbitrary matrix, (56) holds.According to Lemma15, we know that (57) holds if X
22 has the form(59)
X
22
=
T
(
Φ
2
∘
[
T
1
⊤
(
B
2
⊤
D
2
)
Q
1
Ω
+
Ω
Q
1
⊤
(
B
2
⊤
D
2
)
⊤
T
1
]
Ω
-
1
Q
1
⊤
(
B
2
⊤
D
2
)
⊤
T
2
T
2
⊤
(
B
2
⊤
D
2
)
Q
1
Ω
-
1
G
2
)
T
⊤
,where Φ
2
=
(
φ
i
j
)
∈
R
t
×
t, φ
i
j
=
1
/
(
ω
i
2
+
ω
j
2
), 1
≤
i
,
j
≤
t, and G
2
∈
S
R
(
n
-
r
-
t
)
×
(
n
-
r
-
t
) is an arbitrary matrix.From the above discussion, we get the solutions of Problem5.Theorem 17.
GivenA
,
B
∈
R
m
×
n, C
,
D
∈
R
n
×
l. And U
∈
O
R
n
×
n is given by (8). Let the partitions of A
U, B
U, C
⊤
U, and D
⊤
U be as in (32) and (33), respectively. Let the SVDs of (
A
1
⊤
C
1
) and (
A
2
⊤
C
2
) be as (53). Then, the solutions of Problem 5 can be expressed as
(60)
X
=
U
(
X
11
0
0
X
22
)
U
⊤
,
where X
11 and X
22 are given as in (58) and (59).Now, we give another main result of this section. Based on Lemma9 and the properties of Frobenius norm, we have
(61)
∥
A
X
-
B
∥
2
+
∥
X
C
-
D
∥
2
ww
=
∥
A
U
(
0
X
12
-
X
12
⊤
0
)
U
⊤
-
B
∥
2
wwiiii
+
∥
U
(
0
X
12
-
X
12
⊤
0
)
U
⊤
C
-
D
∥
2
ww
=
∥
X
12
A
2
⊤
-
(
-
B
1
⊤
)
∥
2
+
∥
A
1
X
12
-
B
2
∥
2
wwiiii
+
∥
X
12
C
2
-
D
1
∥
2
+
∥
C
1
⊤
X
12
-
(
-
D
2
⊤
)
∥
2
ww
=
∥
(
A
1
C
1
⊤
)
X
12
-
(
B
2
-
D
2
⊤
)
∥
2
wwiiii
+
∥
X
12
(
A
2
⊤
C
2
)
-
(
-
B
1
⊤
D
1
)
∥
2
.
Then, from Lemma 16, the least squares solutions of the matrix equations
(62)
(
A
1
C
1
⊤
)
X
12
=
(
B
2
-
D
2
⊤
)
,
X
12
(
A
2
⊤
C
2
)
=
(
-
B
1
⊤
D
1
)
can be written as
(63)
X
12
=
V
(
Φ
∘
(
V
1
⊤
X
0
T
1
)
(
Σ
-
1
)
2
V
1
⊤
X
0
T
2
V
2
⊤
X
0
T
1
(
Ω
-
1
)
2
G
)
T
⊤
,
where X
0
=
A
1
⊤
B
2
-
B
1
⊤
A
2
-
C
1
D
2
⊤
+
D
1
C
2
⊤, Φ
=
(
φ
i
j
)
∈
R
s
×
t, φ
i
j
=
1
/
(
σ
i
2
+
ω
j
2
), 1
≤
i
≤
s, 1
≤
j
≤
t, and G
∈
R
(
r
-
s
)
×
(
n
-
r
-
t
) is an arbitrary matrix.Theorem 18.
GivenA
,
B
∈
R
m
×
n, C
,
D
∈
R
n
×
l. And U
∈
O
R
n
×
n is given by (8). Let the partitions of A
U, B
U, C
⊤
U, and D
⊤
U be as in (32) and (33), respectively. Let the SVDs of (
A
1
⊤
C
1
) and (
A
2
⊤
C
2
) be as (53). Then, the solutions of Problem 6 can be expressed as
(64)
X
=
U
(
0
X
12
-
X
12
⊤
0
)
U
⊤
,
where X
12 is given as in (63).


## 4. Numerical Examples

In this section, we provide two algorithms to compute the generalized bisymmetric (bi-skew-symmetric) solution and the generalized bisymmetric (bi-skew-symmetric) least squares solution of the matrix equationsA
X
=
B, X
C
=
D and give some examples to illustrate the efficiency of our proposed algorithms.Algorithm 1 (the algorithm about Problems3 and 5).
Step  1. Input A, B, C, D, P.
Step  2. Compute A
1, A
2, B
1, B
2, C
1, C
2, D
1, D
2 by (32) and (33).
Step  3. If any of conditions in (37) and (39) does not hold, then turn to Step 4. Otherwise, compute the generalized bisymmetric solution of the matrix equations A
X
=
B, X
C
=
D by Theorem 13.
Step  4. Compute the generalized bisymmetric least squares solution of the matrix equations A
X
=
B, X
C
=
D by Theorem 17.Algorithm 2 (the algorithm about Problems4 and 6).
Step  1. Input A, B, C, D, P.
Step  2. Compute A
1, A
2, B
1, B
2, C
1, C
2, D
1, D
2 by (32) and (33).
Step  3. If any of conditions in (44) does not hold, then turn to Step 4. Otherwise, compute the generalized bi-skew-symmetric solution of the matrix equations A
X
=
B, X
C
=
D by Theorem 14.
Step  4. Compute the generalized bi-skew-symmetric least squares solution of the matrix equations A
X
=
B, X
C
=
D by Theorem 18.Example 1.
GivenA, B, C, D
∈
R
3
×
3, P
∈
S
O
R
3
×
3 as follows:
(65)
A
=
(
0.9649
0.9572
0.1419
0.1576
0.4854
0.4218
0.9706
0.8003
0.9157
)
,
B
=
(
2.6396
2.2226
-
0.8424
0.2559
0.9047
0.1875
2.4422
1.2061
0.4122
)
,
C
=
(
0.7922
0.0357
0.6787
0.9595
0.8491
0.7577
0.6557
0.9340
0.7431
)
,
D
=
(
1.9731
-
0.3179
1.6446
1.8522
1.4664
1.2772
-
0.0385
0.6894
0.2831
)
,
P
=
(
0.2140
0.9246
-
0.3151
0.9246
-
0.0878
0.3707
-
0.3151
0.3707
0.8737
)
.
Then, there exists
(66)
U
=
(
-
0.6612
-
0.4121
-
0.6269
-
0.6742
-
0.0400
0.7375
-
0.3290
0.9103
-
0.2513
)
∈
O
R
3
×
3
such that
(67)
P
=
U
(
1
0
0
0
1
0
0
0
-
1
)
U
⊤
.
(1) By Algorithm1, the conditions in (37) and (39) hold. Hence, the matrix equations A
X
=
B, X
C
=
D have a generalized bisymmetric solution
(68)
X
G
B
S
*
≈
(
2.9380
-
0.1583
-
0.3089
-
0.1583
2.5979
-
0.7858
-
0.3089
-
0.7858
1.4643
)
.
Furthermore,
(69)
∥
A
X
G
B
S
*
-
B
∥
=
2.2775
e
-
004
,
∥
X
G
B
S
*
C
-
D
∥
=
2.2231
e
-
004
,
∥
X
G
B
S
*
-
X
G
B
S
*
⊤
∥
=
3.5108
e
-
016
,
∥
X
G
B
S
*
-
P
X
G
B
S
*
P
∥
=
7.6271
e
-
004
.
(2) By Algorithm2, at least one of conditions in (44) does not hold. Hence, we get the generalized bi-skew-symmetric least squares solution of the matrix equations A
X
=
B, X
C
=
D, and the generalized bi-skew-symmetric least squares solution is
(70)
X
G
B
S
S
=
(
0
0.2668
-
0.4629
-
0.2668
0
0.4377
0.4629
-
0.4377
0
)
.
Furthermore,
(71)
min
⁡
X
∈
GBSS
R
n
×
n
⁡
∥
A
X
-
B
∥
+
∥
X
C
-
D
∥
=
8.3276
,
∥
X
G
B
S
S
+
X
G
B
S
S
⊤
∥
=
1.2413
e
-
016
,
∥
X
G
B
S
S
+
P
X
G
B
S
S
P
∥
=
1.4017
e
-
004
.Example 2.
GivenA, B, C, D
∈
R
4
×
4, P
∈
S
O
R
4
×
4 as follows:
(72)
A
=
(
0.8147
0.6324
0.9575
0.9572
0.9058
0.0975
0.9649
0.4854
0.1270
0.2785
0.1576
0.8003
0.9134
0.5469
0.9706
0.1419
)
,
B
=
(
0.5972
0.8079
0.2988
-
1.3409
-
0.0105
0.5625
0.5447
-
1.1762
0.6749
0.4230
0.1967
-
0.2930
-
0.0796
0.4102
0.0463
-
1.3852
)
,
C
=
(
0.2769
0.6948
0.4387
0.1869
0.0462
0.3171
0.3816
0.4898
0.0971
0.9502
0.7655
0.4456
0.8235
0.0344
0.7952
0.6463
)
,
D
=
(
-
0.6165
0.2563
-
0.4455
-
0.5208
-
0.3196
-
0.4437
-
0.7539
-
0.5512
-
0.4543
-
0.0793
-
0.2536
0.0091
0.2788
1.1039
0.8516
0.5620
)
,
P
=
(
-
0.0913
-
0.3887
0.8868
0.2326
-
0.3887
0.5024
-
0.0224
0.7720
0.8868
-
0.0224
-
0.0391
0.4598
0.2326
0.7720
0.4598
-
0.3721
)
.
Then, there exists
(73)
U
=
(
-
0.4583
0.4943
-
0.3030
-
0.6737
-
0.4033
-
0.7672
-
0.4944
-
0.0661
-
0.6074
0.3339
-
0.1177
0.7111
-
0.5082
-
0.2359
0.8062
-
0.1899
)
∈
O
R
4
×
4
such that
(74)
P
=
U
(
1
0
0
0
0
1
0
0
0
0
-
1
0
0
0
0
-
1
)
U
⊤
.
(1) By Algorithm1, at least one of conditions in (37) and (39) does not hold. Hence, we get the generalized bisymmetric least squares solution of the matrix equations A
X
=
B, X
C
=
D and the generalized bisymmetric least squares solution is
(75)
X
G
B
S
=
(
0.8056
0.3706
-
0.5480
-
0.3864
0.3706
0.2007
-
0.2672
-
0.2756
-
0.5480
-
0.2672
0.3995
0.1788
-
0.3864
-
0.2756
0.1788
0.2760
)
.
Furthermore,
(76)
min
⁡
X
∈
GBS
R
n
×
n
⁡
∥
A
X
-
B
∥
+
∥
X
C
-
D
∥
=
4.7963
,
∥
X
G
B
S
-
X
G
B
S
⊤
∥
=
1.7988
e
-
016
,
∥
X
G
B
S
-
P
X
G
B
S
P
∥
=
1.4279
e
-
004
.
(2) By Algorithm2, the conditions in (44) hold. Hence, the matrix equations A
X
=
B, X
C
=
D have a generalized bi-skew-symmetric solution
(77)
X
G
B
S
S
*
≈
(
0
-
0.4451
0.4464
-
0.7763
0.4451
0
-
0.7764
-
0.4461
-
0.4464
0.7764
0
-
0.4451
0.7763
0.4461
0.4451
0
)
.
Furthermore,
(78)
∥
A
X
G
B
S
S
*
-
B
∥
=
2.6885
e
-
004
,
∥
X
G
B
S
S
*
C
-
D
∥
=
2.5515
e
-
004
,
∥
X
G
B
S
S
*
+
X
G
B
S
S
*
⊤
∥
=
4.4948
e
-
016
,
∥
X
G
B
S
S
*
+
P
X
G
B
S
S
*
P
∥
=
1.5075
e
-
004
.Example 3.
GivenA
,
B
∈
R
6
×
5, C
,
D
∈
R
5
×
4, and P
∈
S
O
R
5
×
5 as follows:
(79)
A
=
(
0.8147
0.2785
0.9572
0.7922
0.6787
0.9058
0.5469
0.4854
0.9595
0.7577
0.1270
0.9575
0.8003
0.6557
0.7431
0.9134
0.9649
0.1419
0.0357
0.3922
0.6324
0.1576
0.4218
0.8491
0.6555
0.0975
0.9706
0.9157
0.9340
0.1712
)
,
B
=
(
0.7060
0.6948
0.7655
0.7094
0.1190
0.0318
0.3171
0.7952
0.7547
0.4984
0.2769
0.9502
0.1869
0.2760
0.9597
0.0462
0.0344
0.4898
0.6797
0.3404
0.0971
0.4387
0.4456
0.6551
0.5853
0.8235
0.3816
0.6463
0.1626
0.2238
)
,
C
=
(
0.7513
0.9593
0.8407
0.3500
0.2551
0.5472
0.2543
0.1966
0.5060
0.1386
0.8143
0.2511
0.6991
0.1493
0.2435
0.6160
0.8909
0.2575
0.9293
0.4733
)
,
D
=
(
0.3517
0.2858
0.0759
0.1299
0.8308
0.7572
0.0540
0.5688
0.5853
0.7537
0.5308
0.4694
0.5497
0.3804
0.7792
0.0119
0.9172
0.5678
0.9340
0.3371
)
,
P
=
(
-
0.1397
-
0.2600
0.9091
0.2239
0.1904
-
0.2600
0.6869
0.0500
-
0.1319
0.6638
0.9091
0.0500
0.1771
-
0.2921
0.2332
0.2239
-
0.1319
-
0.2921
0.8240
0.4098
0.1904
0.6638
0.2332
0.4098
-
0.5483
)
.
Then, there exists
(80)
U
=
(
-
0.3882
0.4871
0.2054
-
0.0715
-
0.7515
-
0.4756
-
0.3492
-
0.7038
-
0.3709
-
0.1377
-
0.4034
0.6471
-
0.0840
-
0.1630
0.6204
-
0.5101
-
0.4573
0.6653
-
0.2416
0.1720
-
0.4474
-
0.1134
-
0.1130
0.8788
0.0430
)
∈
O
R
4
×
4
such that
(81)
P
=
U
(
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
-
1
0
0
0
0
0
-
1
)
U
⊤
.
(1) By Algorithm1, it is verified that any of conditions in (37) and (39) does not hold. Hence, we get the generalized bisymmetric least squares solution of the matrix equations A
X
=
B, X
C
=
D, and the generalized bisymmetric least squares solution is
(82)
X
G
B
S
=
(
-
0.1363
0.1110
0.3857
0.3289
0.0153
0.1110
0.3641
0.1631
0.2608
0.0069
0.3857
0.1631
-
0.1462
0.2554
0.1547
0.3289
0.2608
0.2554
-
0.0403
0.0532
0.0153
0.0069
0.1547
0.0532
0.6063
)
.
Furthermore,
(83)
min
⁡
X
∈
GBS
R
n
×
n
⁡
∥
A
X
-
B
∥
+
∥
X
C
-
D
∥
=
2.3819
,
∥
X
G
B
S
-
X
G
B
S
⊤
∥
=
1.9192
e
-
016
,
∥
X
G
B
S
-
P
X
G
B
S
P
∥
=
1.9312
e
-
004
.
(2) By Algorithm2, it is verified that any of conditions in (44) does not hold. Hence, we get the generalized bi-skew-symmetric least squares solution of the matrix equations A
X
=
B, X
C
=
D, and the generalized bi-skew-symmetric least squares solution is
(84)
X
G
B
S
S
=
(
0
-
0.0977
-
0.1843
0.2895
-
0.1616
0.0977
0
-
0.2626
-
0.0055
0.1672
0.1843
0.2626
0
-
0.1968
-
0.1138
-
0.2895
0.0055
0.1968
0
0.1110
0.1616
-
0.1672
0.1138
-
0.1110
0
)
.
Furthermore,
(85)
min
⁡
X
∈
GBSS
R
n
×
n
⁡
∥
A
X
-
B
∥
+
∥
X
C
-
D
∥
=
5.3861
,
∥
X
G
B
S
S
+
X
G
B
S
S
⊤
∥
=
1.2889
e
-
016
,
∥
X
G
B
S
S
+
P
X
G
B
S
S
P
∥
=
1.4358
e
-
004
.


## 5. Conclusions

This paper is devoted to considering the generalized bisymmetric (bi-skew-symmetric) solutions of matrix equationsA
X
=
B, X
C
=
D, and the necessary and sufficient conditions for the solvability and the general expression of the solutions are obtained. If the solvability conditions are not satisfied, the generalized bisymmetric (bi-skew-symmetric) least squares solution of the matrix equations is considered. As an auxiliary, two algorithms have been provided to compute the generalized bisymmetric (bi-skew-symmetric) solutions and the generalized bisymmetric (bi-skew-symmetric) least squares solution, and some examples have been given to illustrate that the results are reasonable.



---

*Source: 239465-2014-03-17.xml*