# On the HermitianR-Conjugate Solution of a System of Matrix Equations

Authors: Chang-Zhou Dong; Qing-Wen Wang; Yu-Ping Zhang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

LetR be an n by n nontrivial real symmetric involution matrix, that is,
R=R−1=RT≠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=CandXB=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.


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## Body

## 1. Introduction

Throughout, we denote the complexm×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 ann×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 HermitianR-conjugate solutions to (1.2). We also consider the optimal approximation problem
(1.3)∥X^-E∥=minX∈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∣minX∈RHℂn×n(∥AX-C∥2+∥XB-D∥2)}.
Find X~∈ℂn×n such that
(1.5)∥X~∥=minX∈SL∥X∥.In Section2, 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.R-Conjugate Hermitian Solution to (1.2)

In this section, we establish the solvability conditions and the general expression for the HermitianR-conjugate solution to (1.2).We denoteRℂn×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 matrixR 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 matrixK∈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.
IfK∈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,thatis,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 matrixH∈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.Theorem2.1 implies that an arbitrary complex Hermitian R-conjugate matrix is equivalent to a real symmetric matrix.Lemma 2.2.
For any matrixA∈ℂm×n,A=A1+iA2, where
(2.13)A1=A+A¯2,A2=A-A¯2i.Proof.
For any matrixA∈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,andB2 are real matrix that
(2.15)A1=B1,A2=B2.
Hence, A=A1+iA2 holds, where A1,A2 are defined as (2.13).By Theorem2.1, 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 thatAΓ=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 [7]).
LetA∈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 Lemma2.3, we have the following theorem.Theorem 2.4.
GivenA∈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,andN 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.


## 3. The Solution of Optimal Approximation Problem (1.3)

When the setSX of all Hermitian R-conjugate solution to (1.2) is nonempty, it is easy to verify SX is a closed set. Therefore, the optimal approximation problem (1.3) has a unique solution by [30].Theorem 3.1.
GivenA∈ℂm×n,C∈ℂm×n,B∈ℂn×l,D∈ℂn×l,E∈ℂn×n, and E1=(1/2)(Γ*EΓ+Γ*EΓ¯). Assume SX is nonempty, then the optimal approximation problem (1.3) has a unique solution X^ and
(3.1)X^=Γ(V1M1-1U1TN+V2V2TNTU1M1-1V1T+V2V2TE1V2V2T)Γ*.Proof.
SinceSX is nonempty, X∈SX has the form of (2.27). By Lemma 2.2, Γ*EΓ can be written as
(3.2)Γ*EΓ=E1+iE2,
where
(3.3)E1=12(Γ*EΓ+Γ*EΓ¯),E2=12i(Γ*EΓ-Γ*EΓ¯).
According to (3.2) and the unitary invariance of Frobenius norm
(3.4)∥X-E∥=∥Γ(V1M-1U1TN+V2V2TNTU1M-1V1T+V2GV2T)Γ*-E∥=∥(E1-V1M-1U1TN-V2V2TNTU1M-1V1T-V2GV2T)+iE2∥.
We get
(3.5)∥X-E∥2=∥E1-V1M-1U1TN-V2V2TNTU1M-1V1T-V2GV2T∥2+∥E2∥2.
Then, minX∈SX∥X-E∥ is consistent if and only if there exists G∈ℝ(n-r)×(n-r) such that
(3.6)min∥E1-V1M-1U1TN-V2V2TNTU1M-1V1T-V2GV2T∥.
For the orthogonal matrix V(3.7)∥E1-V1M-1U1TN-V2V2TNTU1M-1V1T-V2GV2T∥2=∥VT(E1-V1M-1U1TN-V2V2TNTU1M-1V1T-V2GV2T)V∥2=∥V1T(E1-V1M-1U1TN)V1∥2+∥V1T(E1-V1M-1U1TN)V2∥2+∥V2T(E1-V2V2TNTU1M-1V1T)V1∥2+∥V2T(E1-V2GV2T)V2∥.
Therefore,
(3.8)min∥E1-V1M-1U1TN-V2V2TNTU1M-1V1T-V2GV2T∥
is equivalent to
(3.9)G=V2TE1V2.
Substituting (3.9) into (2.27), we obtain (3.1).


## 4. The Solution of Problem (1.5)

In this section, we give the explicit expression of the solution to (1.5).Theorem 4.1.
GivenA∈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 (2.25) and system (1.2) has not a solution in HRℂn×n. Then, X∈SL can be expressed as
(4.1)X=ΓV[M1-1U1TNV1M1-1U1TNV2V2TNTU1M1-1Y22]VTΓ*,
where Y22∈ℝ(n-r)×(n-r) is an arbitrary symmetric matrix.Proof.
It yields from (2.17)–(2.21) and (2.25) that
(4.2)∥AX-C∥2+∥XB-D∥2=∥MY-N∥2=∥U[M1000]VTY-N∥2=∥[M1000]VTYV-UTNV∥2.
Assume that
(4.3)VTYV=[Y11Y12Y21Y22],Y11∈ℝr×r,Y22∈ℝ(n-r)×(n-r).
Then, we have
(4.4)∥AX-C∥2+∥XB-D∥2=∥M1Y11-U1TNV1∥2+∥M1Y12-U1TNV2∥2+∥U2TNV1∥2+∥U2TNV2∥2.
Hence,
(4.5)min(∥AX-C∥2+∥XB-D∥2)
is solvable if and only if there exist Y11,Y12 such that
(4.6)∥M1Y11-U1TNV1∥2=min,∥M1Y12-U1TNV2∥2=min.
It follows from (4.6) that
(4.7)Y11=M1-1U1TNV1,Y12=M1-1U1TNV2.
Substituting (4.7) into (4.3) and then into (2.16), we can get that the form of elements in SL is (4.1).Theorem 4.2.
Assume that the notations and conditions are the same as Theorem4.1. Then,
(4.8)∥X~∥=minX∈SL∥X∥
if and only if
(4.9)X~=ΓV[M1-1U1TNV1M1-1U1TNV2V2TNTU1M1-10]VTΓ*.Proof.
In Theorem4.1, it implies from (4.1) that minX∈SL∥X∥ is equivalent to X has the expression (4.1) with Y22=0. Hence, (4.9) holds.


## 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,andN 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.
SupposeA∈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].Example5.2 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.
LetA,B,andC 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].Example5.3 demonstrates that we can get the least squares solution with Algorithm 5.1 when system (1.2) has not Hermitian R-conjugate solutions.



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