On the Hermitian R-Conjugate Solution of a System of Matrix Equations

Let R be an n by n nontrivial real symmetric involution matrix, that is, R R−1 R / In. An n × n complex matrix A is termed R-conjugate if A RAR, where A denotes the conjugate of A. We give necessary and sufficient conditions for the existence of the Hermitian R-conjugate solution to the system of complex matrix equations AX C andXB D and present an expression of the Hermitian R-conjugate solution to this system when the solvability conditions are satisfied. In addition, the solution to an optimal approximation problem is obtained. Furthermore, the least squares Hermitian R-conjugate solution with the least norm to this system mentioned above is considered. The representation of such solution is also derived. Finally, an algorithm and numerical examples are given.


Introduction
Throughout, we denote the complex m × n matrix space by C m×n , the real m × n matrix space by R m×n , and the set of all matrices in R m×n with rank r by R m×n r .The symbols I, A, A T , A * , A † , and A stand for the identity matrix with the appropriate size, the conjugate, the transpose, the conjugate transpose, the Moore-Penrose generalized inverse, and the Frobenius norm of A ∈ C m×n , respectively.We use V n to denote the n × n backward matrix having the elements 1 along the southwest diagonal and with the remaining elements being zeros.
Recall that an n × n complex matrix A is centrohermitian if A V n AV n .Centrohermitian matrices and related matrices, such as k-Hermitian matrices, Hermitian Toeplitz matrices, and generalized centrohermitian matrices, appear in digital signal processing and others areas see, 1-4 .As a generalization of a centrohermitian matrix and related matrices, Trench 5 gave the definition of R-conjugate matrix.A matrix A ∈ C n×n is R-conjugate if A RAR, where R is a nontrivial real symmetric involution matrix, that is, R R −1 R T and R / I n .At the same time, Trench studied the linear equation Az w for R-conjugate matrices in 5 , where z, w are known column vectors.
Investigating the matrix equation with the unknown matrix X being symmetric, reflexive, Hermitian-generalized Hamiltonian, and repositive definite is a very active research topic 6-14 .As a generalization of 1.1 , the classical system of matrix equations has attracted many author's attention.For instance, 15 gave the necessary and sufficient conditions for the consistency of 1.2 , 16, 17 derived an expression for the general solution by using singular value decomposition of a matrix and generalized inverses of matrices, respectively.Moreover, many results have been obtained about the system 1.2 with various constraints, such as bisymmetric, Hermitian, positive semidefinite, reflexive, and generalized reflexive solutions see, 18-28 .To our knowledge, so far there has been little investigation of the Hermitian R-conjugate solution to 1.2 .Motivated by the work mentioned above, we investigate Hermitian R-conjugate solutions to 1.2 .We also consider the optimal approximation problem where E is a given matrix in C n×n and S X the set of all Hermitian R-conjugate solutions to 1.2 .In many cases the system 1.2 has not Hermitian R-conjugate solution.Hence, we need to further study its least squares solution, which can be described as follows: Let RHC n×n denote the set of all Hermitian R-conjugate matrices in C n×n : In Section 2, we present necessary and sufficient conditions for the existence of the Hermitian R-conjugate solution to 1.2 and give an expression of this solution when the solvability conditions are met.In Section 3, we derive an optimal approximation solution to 1.3 .In Section 4, we provide the least squares Hermitian R-conjugate solution to 1.5 .In Section 5, we give an algorithm and a numerical example to illustrate our results.

R-Conjugate Hermitian Solution to 1.2
In this section, we establish the solvability conditions and the general expression for the Hermitian R-conjugate solution to 1.2 .
We denote RC n×n and RHC n×n the set of all R-conjugate matrices and Hermitian Rconjugate matrices, respectively, that is, where R is n × n nontrivial real symmetric involution matrix.Chang et al. in 29 mentioned that for nontrivial symmetric involution matrix R ∈ R n×n , there exist positive integer r and n × n real orthogonal matrix P, Q such that where Throughout this paper, we always assume that the nontrivial symmetric involution matrix R is fixed which is given by 2.2 and 2.3 .Now, we give a criterion of judging a matrix is R-conjugate Hermitian matrix.
which is equivalent to

2.6
Suppose that Substituting 2.7 into 2.6 , we obtain Let Γ P, iQ , and Then, K can be expressed as ΓHΓ * , where Γ is unitary matrix and H is a real matrix.By we obtain H H T .
Conversely, if there exists a symmetric matrix H ∈ R n×n such that K ΓHΓ * , then it follows from 2.3 that 12 that is, K ∈ HRC n×n .
Theorem 2.1 implies that an arbitrary complex Hermitian R-conjugate matrix is equivalent to a real symmetric matrix.

Lemma 2.2. For any matrix
Proof.For any matrix A ∈ C m×n , it is obvious that A A 1 iA 2 , where A 1 , A 2 are defined as 2.13 .Now, we prove that the decomposition

2.14
It follows from A 1 , A 2 , B 1 , and B 2 are real matrix that

2.15
Hence, A A 1 iA 2 holds, where A 1 , A 2 are defined as 2.13 .
By Theorem 2.1, for X ∈ HRC n×n , we may assume that where Γ is defined as 2.4 and 2.17 Then, system 1.2 can be reduced into

has a symmetric solution if and only if
In that case, it has the general solution where G is an arbitrary n − r × n − r symmetric matrix.
By Lemma 2.3, we have the following theorem.

Theorem 2.4. Given
F, G, K, L, M, and N be defined in 2.17 , 2.20 , respectively.Assume that the SV D of M ∈ R 2m 2l ×n is as follows

2.26
In that case, it has the general solution where G is an arbitrary n − r × n − r symmetric matrix.

The Solution of Optimal Approximation Problem 1.3
When the set S X of all Hermitian R-conjugate solution to 1.2 is nonempty, it is easy to verify S X is a closed set.Therefore, the optimal approximation problem 1.3 has a unique solution by 30 .

Theorem 3.1. Given
is nonempty, then the optimal approximation problem 1.3 has a unique solution X and Proof.Since S X is nonempty, X ∈ S X has the form of 2.27 .By Lemma 2.2, Γ * EΓ can be written as where

Journal of Applied Mathematics
According to 3.2 and the unitary invariance of Frobenius norm

3.4
We get

3.6
For the orthogonal matrix

The Solution of Problem 1.5
In this section, we give the explicit expression of the solution to 1.5 .

Theorem 4.1. Given
and N be defined in 2.17 , 2.20 , respectively.Assume that the SV D of M ∈ R 2m 2l ×n is as 2.25 and system 1.2 has not a solution in HRC n×n .Then, X ∈ S L can be expressed as where Y 22 ∈ R n−r × n−r is an arbitrary symmetric matrix.
Proof.It yields from 2.17 -2.21 and 2.25 that

4.2
Assume that Then, we have

4.4
Hence, Proof.In Theorem 4.1, it implies from 4.1 that min X∈S L X is equivalent to X has the expression 4.1 with Y 22 0. Hence, 4.9 holds.

An Algorithm and Numerical Example
Base on the main results of this paper, we in this section propose an algorithm for finding the solution of the approximation problem 1.3 and the least squares problem with least norm 1.5 .All the tests are performed by MATLAB 6.5 which has a machine precision of around 10 −16 .
F, G, K, L, M, and N by 2.17 and 2.20 .
3 Compute the singular value decomposition of M with the form of 2.25 .4 If 2.26 holds, then input E ∈ C n×n and compute the solution X of problem 1.3 according 3.1 , else compute the solution X to problem 1.5 by 4.9 .
To show our algorithm is feasible, we give two numerical example.Let an nontrivial symmetric involution be We obtain P, Q in 2.2 by using the spectral decomposition of R, then by 2.4  We can verify that 2.26 does not hold.By Algorithm 5.
that we can solve the optimal approximation problem with Algorithm 5.1 when system 1.2 have Hermitian R-conjugate solutions.Example 5.3.Let A, B, and C be the same as Example 5.2, and let D in Example 5.2 be changed into D 0.1872i −0.0009 0.1756i 1.6746 0.0494i 0.0024 0.2704i 0.1775 0.4194i 0.7359 − 0.6189i −0.0548 0.3444i 0.0093 − 0.3075i −0.4731 − 0.1636i 0.0337i 0.1209 − 0.1864i −0.2484 − 3.8817i Example 5.3 demonstrates that we can get the least squares solution with Algorithm 5.1 when system 1.2 has not Hermitian R-conjugate solutions.