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

and Applied Analysis 3 Conversely, for any X 11 ∈ SR and X 22 ∈ SR, it is easy to verify thatX = X. Using (8), we have PXP = 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 ⊤ = U( X 11 0 0 X 22 )U ⊤


Introduction
The class of matrix equations, namely,  = ,  = , where , , , and  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., ).
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  with the constraint  = , where  is a given Hermitian matrix satisfying  2 =  and  = ±1.
In this paper,  × ,  × ,  × , and  × denote the set of all  ×  real matrices, the set of all  ×  real orthogonal matrices, the set of all  ×  real symmetric matrices, and the set of all  ×  real symmetric orthogonal matrices, respectively. ⊤ represents the transpose of the real matrix  and ‖ ⋅ ‖ stands for the Frobenius norm induced by the inner product.( ) denotes a row block matrix and ∘ denotes the Hadamard product produced by  and , namely, ∘ = (    ).The symbol   stands for the identity matrix of order .Let  † be the Moore-Penrose generalized inverse of a matrix  ∈  × , which is defined to be the unique solution  ∈  × satisfying the following four matrix equations: If  =  = 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 Section 2, 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.

The Generalized Bisymmetric (Bi-Skew-Symmetric) Solutions of the Matrix Equations 𝐴𝑋 = 𝐵, 𝑋𝐶 = 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 Problems 3 and 4.
Proof.For  ∈  × , by Definition 2, we have By ( 16) and Lemma 7, we obtain Let  12 =  ⊤ 1  2 and  21 =  ⊤ 2  1 ; it is easy to verify that  21 = − ⊤ 12 .And we have Conversely, for any  12 ∈  ×(−) , it is easy to verify that  = − ⊤ .Using (8), we have This implies that Remark 10.For Lemma 8, 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 that Then the system of matrix equations is consistent if and only if in which case, the general solutions of the system can be expressed as where  is an arbitrary real matrix with compatible dimension.
Then, the following statements are equivalent.
(i) The matrix equations have a solution  ∈  × .
(ii) The system of matrix equations have a solution  ∈  × ; in this case, the symmetric solution of the matrix equations (24) is the symmetric solutions of the matrix equations (24) can be expressed as where  ∈  × is an arbitrary matrix.
Proof.(i) ⇔ (ii).It is not difficult to get that (i) is equivalent to (ii).Further, if  is a solution of the matrix equations (25), then Moreover, 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 Substituting ( 31) into ( 26) yields (28).The proof is completed.
For  ∈  × which is given by ( 8), partition From Lemma 8, we know that the matrix equations  = ,  =  have a solution  ∈  × if and only if there exist  11 ∈  × and  22 ∈  (−)×(−) such that that is, Let By Lemma 12, the matrix equations (35) have a solution  11 ∈  × if and only if in which case the general solutions can be expressed as where  1 ∈  × is an arbitrary matrix.
Similarly, the matrix equations (36) have a solution  22 ∈  (−)×(−) if and only if in which case the general solutions can be expressed as where  2 ∈  (−)×(−) is an arbitrary matrix.Now, based on the above discussion, we give the solvability conditions and the general expression of the solutions of Problem 3.
From Lemma 9, we know that the matrix equations  = ,  =  have a solution  ∈  × if and only if there exists  12 ∈  ×(−) such that that is, . By Lemma 11, the system of matrix equations (43) has a solution  12 ∈  ×(−) if and only if in which case the general solutions can be expressed as where  ∈  ×(−) is an arbitrary matrix.Therefore, we have the following result about Problem 4.
Theorem 14.Given ,  ∈  × , ,  ∈  × .And  ∈  × is given by (8).Let the partitions of , ,  ⊤ , and  ⊤  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 where  12 is given as in (45).

The Generalized Bisymmetric (Bi-Skew-Symmetric) Least Squares Solutions of the Matrix Equations 𝐴𝑋=𝐵, 𝑋𝐶=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 Problems 5 and 6.Firstly, we present some lemmas which will play important roles in the following.
From the above discussion, we get the solutions of Problem 5.

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 equations  = ,  =  and give some examples to illustrate the efficiency of our proposed algorithms.
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  = ,  =  by Theorem 13.
Step 4. Compute the generalized bisymmetric least squares solution of the matrix equations  = ,  =  by Theorem 17.