1. Introduction
Throughout, we denote the complex m×n matrix space by ℂm×n, the real m×n matrix space by ℝm×n, and the set of all matrices in ℝm×n with rank r by ℝrm×n. The symbols I,A¯,AT,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∈ℂm×n, respectively. We use Vn 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¯=VnAVn. 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∈ℂn×n is R-conjugate if A¯=RAR, where R is a nontrivial real symmetric involution matrix, that is, R=R-1=RT and R≠In. 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
(1.1) AX=B
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
(1.2)AX=C, XB=D
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
(1.3)∥X^-E∥=min X∈SX∥X-E∥,
where E is a given matrix in ℂn×n and SX 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 RHℂn×n denote the set of all Hermitian R-conjugate matrices in ℂn×n:
(1.4)SL={X∣min X∈RHℂn×n(∥AX-C∥2+∥XB-D∥2)}.
Find X~∈ℂn×n such that
(1.5)∥X~∥=min X∈SL∥X∥.

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.

2. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M61"><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:math></inline-formula>-Conjugate Hermitian Solution to (<xref ref-type="disp-formula" rid="EEq1.2">1.2</xref>)
In this section, we establish the solvability conditions and the general expression for the Hermitian R-conjugate solution to (1.2).

We denote Rℂn×n and RHℂn×n the set of all R-conjugate matrices and Hermitian R-conjugate matrices, respectively, that is,
(2.1)Rℂn×n={A∣A¯=RAR},HRℂn×n={A∣A¯=RAR,A=A*},
where R is n×n nontrivial real symmetric involution matrix.

Chang et al. in [29] mentioned that for nontrivial symmetric involution matrix R∈ℝn×n, there exist positive integer r and n×n real orthogonal matrix [P, Q] such that
(2.2)R=[P,Q][Ir00-In-r][PTQT],
where P∈ℝn×r, Q∈ℝn×(n-r). By (2.2),
(2.3)RP=P, RQ=-Q, PTP=Ir, QTQ=In-r, PTQ=0, QTP=0.

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.

Theorem 2.1.
A matrix K∈HRℂn×n if and only if there exists a symmetric matrix H∈ℝn×n such that K=ΓHΓ*, where
(2.4)Γ=[P,iQ],
with P,Q being the same as (2.2).

Proof.
If K∈HRℂn×n, then K¯=RKR. By (2.2),
(2.5)K¯=RKR=[P,Q][Ir00-In-r][PTQT]K[P,Q][Ir00-In-r][PTQT],
which is equivalent to
(2.6)[PTQT]K¯[P,Q] =[Ir00-In-r][PTQT]K[P,Q][Ir00-In-r].
Suppose that
(2.7)[PTQT]K[P,Q]=[K11K12K21K22].
Substituting (2.7) into (2.6), we obtain
(2.8)[K11¯K12¯K21¯K22¯]=[Ir00-In-r][K11K12K21K22][Ir00-In-r]=[K11-K12-K21K22].
Hence, K11¯=K11, K12¯=-K12, K21¯=-K21, K22¯=K22, that is, K11, iK12, iK21, K22 are real matrices. If we denote M=iK12, N=-iK21, then by (2.7)
(2.9)K=[P,Q][K11K12K21K22][PTQT]=[P,iQ][K11MNK22][PT-iQT].
Let Γ=[P, iQ], and
(2.10)H=[K11MNK22].
Then, K can be expressed as ΓHΓ*, where Γ is unitary matrix and H is a real matrix. By K=K*(2.11)ΓHTΓ*=K*=K=ΓHΓ*,
we obtain H=HT.

Conversely, if there exists a symmetric matrix H∈ℝn×n such that K=ΓHΓ*, then it follows from (2.3) that
(2.12)RKR=RΓHΓ*R=R[P,iQ]H[PT-iQT]R=[P,-iQ]H[PTiQT]=Γ¯HΓ*¯=K¯,K*=ΓHTΓ*=ΓHΓ*=K,
that is, K∈HRℂn×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 A∈ℂm×n, A=A1+iA2, where
(2.13)A1=A+A¯2, A2=A-A¯2i.

Proof.
For any matrix A∈ℂm×n, it is obvious that A=A1+iA2, where A1, A2 are defined as (2.13). Now, we prove that the decomposition A=A1+iA2 is unique. If there exist B1, B2 such that A=B1+iB2, then
(2.14)A1-B1+i(B2-A2)=0.
It follows from A1,A2,B1, and B2 are real matrix that
(2.15)A1=B1, A2=B2.
Hence, A=A1+iA2 holds, where A1, A2 are defined as (2.13).

By Theorem 2.1, for X∈HRℂn×n, we may assume that
(2.16)X=ΓYΓ*,
where Γ is defined as (2.4) and Y∈ℝn×n is a symmetric matrix.

Suppose that AΓ=A1+iA2∈ℂm×n, CΓ=C1+iC2∈ℂm×n, Γ*B=B1+iB2∈ℂn×l, and Γ*D=D1+iD2∈ℂn×l, where
(2.17)A1=AΓ+AΓ¯2, A2=AΓ-AΓ¯2i, C1=CΓ+CΓ¯2, C2=CΓ-CΓ¯2i,B1=Γ*B+Γ*B¯2, B2=Γ*B-Γ*B¯2i, D1=Γ*D+Γ*D¯2, D2=Γ*D-Γ*D¯2i.
Then, system (1.2) can be reduced into
(2.18)(A1+iA2)Y=C1+iC2, Y(B1+iB2)=D1+iD2,
which implies that
(2.19)[A1A2]Y=[C1C2], Y[B1,B2]=[D1,D2].
Let
(2.20)F=[A1A2], G=[C1C2], K=[B1,B2],L=[D1,D2], M=[FKT], N=[GLT].
Then, system (1.2) has a solution X in HRℂn×n if and only if the real system
(2.21)MY=N
has a symmetric solution Y in ℝn×n.

Lemma 2.3 (Theorem 1 in [<xref ref-type="bibr" rid="B7">7</xref>]).
Let A∈ℝm×n. The SVD of matrix A is as follows
(2.22)A=U[Σ000]VT,
where U=[U1, U2]∈ℝm×m and V=[V1, V2]∈ℝn×n are orthogonal matrices, Σ=diag (σ1,…,σr), σi>0 (i=1,…,r), r=rank (A), U1∈ℝm×r, V1∈ℝn×r. Then, (1.1) has a symmetric solution if and only if
(2.23)ABT=BAT, U2TB=0.
In that case, it has the general solution
(2.24)X=V1Σ-1U1TB+V2V2TBTU1Σ-1V1T+V2GV2T,
where G is an arbitrary (n-r)×(n-r) symmetric matrix.

By Lemma 2.3, we have the following theorem.

Theorem 2.4.
Given A∈ℂm×n, C∈ℂm×n, B∈ℂn×l, and D∈ℂn×l. Let A1, A2, C1, C2, B1, B2, D1, D2, F, G, K, L, M, and N be defined in (2.17), (2.20), respectively. Assume that the SVD of M∈ℝ(2m+2l)×n is as follows
(2.25)M=U[M1000]VT,
where U=[U1, U2]∈ℝ(2m+2l)×(2m+2l) and V=[V1, V2]∈ℝn×n are orthogonal matrices, M1=diag (σ1,…,σr), σi>0 (i=1,…,r), r=rank (M), U1∈ℝ(2m+2l)×r, V1∈ℝn×r. Then, system (1.2) has a solution in HRℂn×n if and only if
(2.26)MNT=NMT, U2TN=0.
In that case, it has the general solution
(2.27)X=Γ(V1M1-1U1TN+V2V2TNTU1M1-1V1T+V2GV2T)Γ*,
where G is an arbitrary (n-r)×(n-r) symmetric matrix.

5. 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.

Algorithm 5.1.
(
1
)
Input A∈ℂm×n, C∈ℂm×n, B∈ℂn×l, D∈ℂn×l.

(
2
)
Compute A1, A2, C1, C2, B1, B2, D1, D2, 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∈ℂn×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
(5.1)R=[10000-1000010000-1].
We obtain [P, Q] in (2.2) by using the spectral decomposition of R, then by (2.4)
(5.2)Γ=[100000i00100000i].

Example 5.2.
Suppose A∈ℂ2×4, C∈ℂ2×4, B∈ℂ4×3, D∈ℂ4×3, and
(5.3)A=[3.33-5.987i45i7.21-i0-0.66i7.6941.123i],C=[0.2679-0.0934i0.0012+4.0762i-0.0777-0.1718i-1.2801i0.2207-0.1197i0.08770.7058i],B=[4+12i2.369i4.256-5.111i4i4.66i8.21-5i04.83i56+i2.22i-4.6667i],D=[0.0616+0.1872i-0.0009+0.1756i1.6746-0.0494i0.0024+0.2704i0.1775+0.4194i0.7359-0.6189i-0.0548+0.3444i0.0093-0.3075i-0.4731-0.1636i0.0337i0.1209-0.1864i-0.2484-3.8817i].
We can verify that (2.26) holds. Hence, system (1.2) has an Hermitian R-conjugate solution. Given
(5.4)E=[7.35i8.389i99.256-6.51i-4.6i1.554.56i7.71-7.5ii5i0-4.556i-7.994.22i05.1i0].
Applying Algorithm 5.1, we obtain the following:
(5.5)X^=[1.55970.0194i2.87050.0002i-0.0194i9.00010.2005i-3.99972.8705-0.2005i-0.04527.9993i-0.0002i-3.9997-7.9993i5.6846].

Example 5.2 illustrates 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
(5.6)D=[0.0616+0.1872i-0.0009+0.1756i1.6746+0.0494i0.0024+0.2704i0.1775+0.4194i0.7359-0.6189i-0.0548+0.3444i0.0093-0.3075i-0.4731-0.1636i0.0337i0.1209-0.1864i-0.2484-3.8817i].
We can verify that (2.26) does not hold. By Algorithm 5.1, we get
(5.7)X~=[0.522.2417i0.49140.3991i-2.2417i8.66340.1921i-2.82320.4914-0.1921i0.14061.3154i-0.3991i-2.8232-1.3154i6.3974].

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.