# Constrained Solutions of a System of Matrix Equations

Authors: Qing-Wen Wang; Juan Yu
Journal: Journal of Applied Mathematics
 (2012)

---

## Abstract

We derive the necessary and sufficient conditions of and the expressions for the orthogonal solutions, the symmetric orthogonal solutions, and the skew-symmetric orthogonal solutions of the system of matrix equationsAX=B and XC=D, respectively. When the matrix equations are not consistent, the least squares symmetric orthogonal solutions and the least squares skew-symmetric orthogonal solutions are respectively given. As an auxiliary, an algorithm is provided to compute the least squares symmetric orthogonal solutions, and meanwhile an example is presented to show that it is reasonable.


---

## Body

## 1. Introduction

Throughout this paper, the following notations will be used.ℝm×n, Oℝn×n, Sℝn×n, and ASℝn×n denote the set of all m×n real matrices, the set of all n×n orthogonal matrices, the set of all n×n symmetric matrices, and the set of all n×n skew-symmetric matrices, respectively. In is the identity matrix of order n. (·)T and tr(·) represent the transpose and the trace of the real matrix, respectively. ∥·∥ stands for the Frobenius norm induced by the inner product. The following two definitions will also be used.Definition 1.1 (see [1]).
A real matrixX∈ℝn×n is said to be a symmetric orthogonal matrix if XT=X and XTX=In.Definition 1.2 (see [2]).
A real matrixX∈ℝ2m×2m is called a skew-symmetric orthogonal matrix if XT=-X and XTX=In.The set of alln×n symmetric orthogonal matrices and the set of all 2m×2m skew-symmetric orthogonal matrices are, respectively, denoted by SOℝn×n and SSOℝ2m×2m. Since the linear matrix equation(s) and its optimal approximation problem have great applications in structural design, biology, control theory, and linear optimal control, and so forth, see, for example, [3–5], there has been much attention paid to the linear matrix equation(s). The well-known system of matrix equations
(1.1)AX=B,XC=D,
as one kind of linear matrix equations, has been investigated by many authors, and a series of important and useful results have been obtained. For instance, the system (1.1) with unknown matrix X being bisymmetric, centrosymmetric, bisymmetric nonnegative definite, Hermitian and nonnegative definite, and (P,Q)-(skew) symmetric has been, respectively, investigated by Wang et al. [6, 7], Khatri and Mitra [8], and Zhang and Wang [9]. Of course, if the solvability conditions of system (1.1) are not satisfied, we may consider its least squares solution. For example, Li et al. [10] presented the least squares mirrorsymmetric solution. Yuan [11] got the least-squares solution. Some results concerning the system (1.1) can also be found in [12–18].Symmetric orthogonal matrices and skew-symmetric orthogonal matrices play important roles in numerical analysis and numerical solutions of partial differential equations. Papers [1, 2], respectively, derived the symmetric orthogonal solution X of the matrix equation XC=D and the skew-symmetric orthogonal solution X of the matrix equation AX=B. Motivated by the work mentioned above, we in this paper will, respectively, study the orthogonal solutions, symmetric orthogonal solutions, and skew-symmetric orthogonal solutions of the system (1.1). Furthermore, if the solvability conditions are not satisfied, the least squares skew-symmetric orthogonal solutions and the least squares symmetric orthogonal solutions of the system (1.1) will be also given.The remainder of this paper is arranged as follows. In Section2, some lemmas are provided to give the main results of this paper. In Sections 3, 4, and 5, the necessary and sufficient conditions of and the expression for the orthogonal, the symmetric orthogonal, and the skew-symmetric orthogonal solutions of the system (1.1) are, respectively, obtained. In Section 6, the least squares skew-symmetric orthogonal solutions and the least squares symmetric orthogonal solutions of the system (1.1) are presented, respectively. In addition, an algorithm is provided to compute the least squares symmetric orthogonal solutions, and meanwhile an example is presented to show that it is reasonable. Finally, in Section 7, some concluding remarks are given.


## 2. Preliminaries

In this section, we will recall some lemmas and the specialC-S decomposition which will be used to get the main results of this paper.Lemma 2.1 (see [1, Lemmas 1 and 2]).
GivenC∈ℝ2m×n,  D∈ℝ2m×n. The matrix equation YC=D has a solution Y∈Oℝ2m×2m if and only if DTD=CTC. Let the singular value decompositions of C and D be, respectively,
(2.1)C=U(Π000)VT,D=W(Π000)VT,
where
(2.2)Π=diag(σ1,…,σk)>0,k=rank(C)=rank(D),U=(U1U2)∈Oℝ2m×2m,U1∈ℝ2m×k,W=(W1W2)∈Oℝ2m×2m,W1∈ℝ2m×k,V=(V1V2)∈Oℝn×n,V1∈ℝn×k.
Then the orthogonal solutions of YC=D can be described as
(2.3)Y=W(Ik00P)UT,
where P∈Oℝ(2m-k)×(2m-k) is arbitrary.Lemma 2.2 (see [2, Lemmas 1 and 2]).
GivenA∈ℝn×m,  B∈ℝn×m. The matrix equation AX=B has a solution X∈Oℝm×m if and only if AAT=BBT. Let the singular value decompositions of A and B be, respectively,
(2.4)A=U(Σ000)VT,B=U(Σ000)QT,
where
(2.5)Σ=diag(δ1,…,δl)>0,l=rank(A)=rank(B),U=(U1U2)∈Oℝn×n,U1∈ℝn×l,V=(V1V2)∈Oℝm×m,V1∈ℝm×l,Q=(Q1Q2)∈Oℝm×m,Q1∈ℝm×l.
Then the orthogonal solutions of AX=B can be described as
(2.6)X=V(Il00W)QT,
where W∈Oℝ(m-l)×(m-l) is arbitrary.Lemma 2.3 (see [2, Theorem 1]).
If(2.7)X=(X11X12X21X22)∈Oℝ2m×2m,X11∈ASℝk×k,
then the C-S decomposition of X can be expressed as
(2.8)(D100D2)T(X11X12X21X22)(D100R2)=(Σ11Σ12Σ21Σ22),
where D1∈Oℝk×k,D2,R2∈Oℝ(2m-k)×(2m-k),
(2.9)Σ11=(I~000C~0000),Σ12=(0000S000I),Σ21=(0000S000I),Σ22=(I000-C~T0000),I~=diag(I~1,…,I~r1),I~1=⋯=I~r1=(01-10),C~=diag(C1,…,Cl1),Ci=(0ci-ci0),i=1,…,l1;S=diag(S1,…,Sl1),Si=(si00si),SiTSi+CiTCi=I2,i=1,…,l1;2r1+2l1=rank(X11),Σ21=Σ12T,k-2r1=rank(X21).Lemma 2.4 (see [1, Theorem 1]).
If(2.10)K=(K11K12K21K22)∈Oℝm×m,K11∈Sℝl×l,
then the C-S decomposition of K can be described as
(2.11)(D100D2)T(K11K12K21K22)(D100R2)=(Π11Π12Π21Π22),
where D1∈Oℝl×l,  D2,  R2∈Oℝ(m-l)×(m-l),
(2.12)Π11=(I~000C~0000),Π12=(0000S000I),Π21=(0000S000I),Π22=(I000-C~0000),I~=diag(i1,…,ir),ij=±1,j=1,…,r;S=diag(s1′,…,sl′),C~=diag(c1′,…,cl′),si′2=1-ci′2,i=1′,…,l′;r+l′=rank(K11),Π21=Π12T.Remarks 2.5.
In order to know theC-S decomposition of an orthogonal matrix with a k×k leading (skew-) symmetric submatrix for details, one can deeply study the proof of Theorem  1 in [1] and [2].Lemma 2.6.
GivenA∈ℝn×m,  B∈ℝn×m. Then the matrix equation AX=B has a solution X∈SOℝm×m if and only if AAT=BBT and ABT=BAT. When these conditions are satisfied, the general symmetric orthogonal solutions can be expressed as
(2.13)X=V~(I2l-r00G)Q~T,
where
(2.14)Q~=JQdiag(I,D2)∈Oℝm×m,V~=JVdiag(I,R2)∈Oℝm×m,J=(I0000I0I0),
and G∈SOℝ(m-2l+r)×(m-2l+r) is arbitrary.Proof.
The Necessity. Assume X∈SOℝm×m is a solution of the matrix equation AX=B, then we have
(2.15)BBT=AXXTAT=AAT,BAT=AXAT=AXTAT=ABT.
The Sufficiency. Since the equality AAT=BBT holds, then by Lemma 2.2, the singular value decompositions of A and B can be, respectively, expressed as (2.4). Moreover, the condition ABT=BAT means
(2.16)U(Σ000)VTQ(Σ000)UT=U(Σ000)QTV(Σ000)UT,
which can be written as
(2.17)V1TQ1=Q1TV1.
From Lemma 2.2, the orthogonal solutions of the matrix equation AX=B can be described as (2.6). Now we aim to find that X in (2.6) is also symmetric. Suppose that X is symmetric, then we have
(2.18)(Il00WT)VTQ=QTV(Il00W),
together with the partitions of the matrices Q and V in Lemma 2.2, we get
(2.19)V1TQ1=Q1TV1,(2.20)Q1TV2W=V1TQ2,(2.21)Q2TV2W=WTV2TQ2.
By (2.17), we can get (2.19). Now we aim to find the orthogonal solutions of the system of matrix equations (2.20) and (2.21). Firstly, we obtain from (2.20) that Q1TV2(Q1TV2)T=V1TQ2(V1TQ2)T, then by Lemma 2.2, (2.20) has an orthogonal solution W. By (2.17), the l×l leading principal submatrix of the orthogonal matrix VTQ is symmetric. Then we have, from Lemma 2.4,
(2.22)V1TQ2=D1Π12R2T,(2.23)Q1TV2=D1Π12D2T,(2.24)V2TQ2=D2Π22R2T.
From (2.20), (2.22), and (2.23), the orthogonal solution W of (2.20) is
(2.25)W=D2(G000I000I)R2T,
where G∈Oℝ(m-2l+r)×(m-2l+r) is arbitrary. Combining (2.21), (2.24), and (2.25) yields GT=G,thatis,G is a symmetric orthogonal matrix. Denote
(2.26)V^=Vdiag(I,D2),Q^=Qdiag(I,R2),
then the symmetric orthogonal solutions of the matrix equation AX=B can be expressed as
(2.27)X=V^(I000G000I)Q^T.
Let the partition matrix J be
(2.28)J=(I0000I0I0),
compatible with the block matrix
(2.29)(I000G000I).
Put
(2.30)V~=JV^,Q~=JQ^,
then the symmetric orthogonal solutions of the matrix equation AX=B can be described as (2.13).SettingA=CT,  B=DT, and X=YT in [2, Theorem 2], and then by Lemmas 2.1 and 2.3, we can have the following result.Lemma 2.7.
GivenC∈ℝ2m×n,  D∈ℝ2m×n. Then the equation has a solution Y∈SSOℝ2m×2m if and only if DTD=CTC and DTC=-CTD. When these conditions are satisfied, the skew-symmetric orthogonal solutions of the matrix equation YC=D can be described as
(2.31)Y=W~(I00H)U~T,
where
(2.32)W~=J′Wdiag(I,-D2)∈Oℝ2m×2m,U~=J′Udiag(I,R2)∈Oℝ2m×2m,J′=(I0000I0I0),
and H∈SSOℝ2r×2r is arbitrary.


## 3. The Orthogonal Solutions of the System (1.1)

The following theorems give the orthogonal solutions of the system (1.1).Theorem 3.1.
GivenA,  B∈ℝn×m and C,  D∈ℝm×n, suppose the singular value decompositions of A and B are, respectively, as (2.4). Denote
(3.1)QTC=(C1C2),VTD=(D1D2),
where C1, D1∈ℝl×n, and C2,D2∈ℝ(m-l)×n. Let the singular value decompositions of C2 and D2 be, respectively,
(3.2)C2=U~(Π000)V~T,D2=W~(Π000)V~T,
where U~,  W~∈Oℝ(m-l)×(m-l),  V~∈Oℝn×n,  Π∈ℝk×k is diagonal, whose diagonal elements are nonzero singular values of C2 or D2. Then the system (1.1) has orthogonal solutions if and only if
(3.3)AAT=BBT,C1=D1,D2TD2=C2TC2.
In which case, the orthogonal solutions can be expressed as
(3.4)X=V^(Ik+l00G′)Q^T,
where
(3.5)V^=V(Il00W~)∈Oℝm×m,Q^=Q(Il00U~)∈Oℝm×m,
and G′∈Oℝ(m-k-l)×(m-k-l) is arbitrary.Proof.
Let the singular value decompositions ofA and B be, respectively, as (2.4). Since the matrix equation AX=B has orthogonal solutions if and only if
(3.6)AAT=BBT,
then by Lemma 2.2, its orthogonal solutions can be expressed as (2.6). Substituting (2.6) and (3.1) into the matrix equation XC=D, we have C1=D1 and WC2=D2. By Lemma 2.1, the matrix equation WC2=D2 has orthogonal solution W if and only if
(3.7)D2TD2=C2TC2.
Let the singular value decompositions of C2 and D2 be, respectively,
(3.8)C2=U~(Π000)V~T,D2=W~(Π000)V~T,
where U~,  W~∈Oℝ(m-l)×(m-l),  V~∈Oℝn×n,  Π∈ℝk×k is diagonal, whose diagonal elements are nonzero singular values of C2 or D2. Then the orthogonal solutions can be described as
(3.9)W=W~(Ik00G′)U~T,
where G′∈Oℝ(m-k-l)×(m-k-l) is arbitrary. Therefore, the common orthogonal solutions of the system (1.1) can be expressed as
(3.10)X=V(Il00W)QT=V(Il00W~)(Il000Ik000G′)(Il00U~T)QT=V^(Ik+l00G′)Q^T,
where
(3.11)V^=V(Il00W~)∈Oℝm×m,Q^=Q(Il00U~)∈Oℝm×m,
and G′∈Oℝ(m-k-l)×(m-k-l) is arbitrary.The following theorem can be shown similarly.Theorem 3.2.
GivenA,  B∈ℝn×m and C,  D∈ℝm×n, let the singular value decompositions of C and D be, respectively, as (2.1). Partition
(3.12)AW=(A1A2),BU=(B1B2),
where A1,B1∈ℝn×k,  A2,B2∈ℝn×(m-k). Assume the singular value decompositions of A2 and B2 are, respectively,
(3.13)A2=U~(Σ000)V~T,B2=U~(Σ000)Q~T,
where V~,  Q~∈Oℝ(m-k)×(m-k),  U~∈Oℝn×n,  Σ∈ℝl′×l′is diagonal, whose diagonal elements are nonzero singular values of A2 or B2. Then the system (1.1) has orthogonal solutions if and only if
(3.14)DTD=CTC,A1=B1,A2A2T=B2B2T.
In which case, the orthogonal solutions can be expressed as
(3.15)X=W^(Ik+l′00H′)U^T,
where
(3.16)W^=W(Ik00W~)∈Oℝm×m,U^=U(Ik00U~)∈Oℝm×m,
and H′∈Oℝ(m-k-l′)×(m-k-l′) is arbitrary.


## 4. The Symmetric Orthogonal Solutions of the System (1.1)

We now present the symmetric orthogonal solutions of the system (1.1).Theorem 4.1.
GivenA,B∈ℝn×m,  C,D∈ℝm×n. Let the symmetric orthogonal solutions of the matrix equation AX=B be described as in Lemma 2.6. Partition
(4.1)Q~TC=(C1′C2′),V~TD=(D1′D2′),
where C1′,D1′∈ℝ(2l-r)×n,  C2′,D2′∈ℝ(m-2l+r)×n. Then the system (1.1) has symmetric orthogonal solutions if and only if
(4.2)AAT=BBT,ABT=BAT,C1′=D1′,D2′TD2′=C2′TC2′,D2′TC2′=C2′TD2′.
In which case, the solutions can be expressed as
(4.3)X=V^(I00G′′)Q^T,
where
(4.4)V^=V(I2l-r00W~)∈Oℝm×m,Q^=Q(I2l-r00U~)∈Oℝm×m,
and G′′∈SOℝ(m-2l+r-2l′+r′)×(m-2l+r-2l′+r′) is arbitrary.Proof.
From Lemma2.6, we obtain that the matrix equation AX=B has symmetric orthogonal solutions if and only if AAT=BBT and ABT=BAT. When these conditions are satisfied, the general symmetric orthogonal solutions can be expressed as
(4.5)X=V~(I2l-r00G)Q~T,
where G∈SOℝ(m-2l+r)×(m-2l+r) is arbitrary, Q~∈Oℝm×m,  V~∈Oℝm×m. Inserting (4.1) and (4.5) into the matrix equation XC=D, we get C1′=D1′ and GC2′=D2′. By [1, Theorem  2], the matrix equation GC2′=D2′ has a symmetric orthogonal solution if and only if
(4.6)D2′TD2′=C2′TC2′,D2′TC2′=C2′TD2′.
In which case, the solutions can be described as
(4.7)G=W~(I2l′-r′00G′′)U~T,
where G′′∈SOℝ(m-2l+r-2l′+r′)×(m-2l+r-2l′+r′) is arbitrary, W~∈Oℝ(m-2l+r)×(m-2l+r), and U~∈Oℝ(m-2l+r)×(m-2l+r). Hence the system (1.1) has symmetric orthogonal solutions if and only if all equalities in (4.2) hold. In which case, the solutions can be expressed as
(4.8)X=V~(I2l-r00G)Q~T=V~(I2l-r00W~)(I2l-r00(I00G′′))(I2l-r00U~T)Q~T,
that is, the expression in (4.3).The following theorem can also be obtained by the method used in the proof of Theorem4.1.Theorem 4.2.
GivenA,B∈ℝn×m,  C,D∈ℝm×n. Let the symmetric orthogonal solutions of the matrix equation XC=D be described as
(4.9)X=M~(I2k-r00G)N~T,
where M~,  N~∈Oℝm×m,  G∈SOℝ(m-2k+r)×(m-2k+r). Partition
(4.10)AM~=(M1M2),BN~=(N1N2),M1,N1∈ℝn×(2k-r),M2,N2∈ℝn×(m-2k+r).
Then the system (1.1) has symmetric orthogonal solutions if and only if
(4.11)DTD=CTC,DTC=CTD,M1=N1,M2M2T=N2N2T,M2N2T=N2M2T.
In which case, the solutions can be expressed as
(4.12)X=M^(I00H′′)N^T,
where
(4.13)M^=M~(I2k-r00W~1)∈Oℝm×m,N^=N~(I2k-r00U~1)∈Oℝm×m,
and H′′∈SOℝ(m-2k+r-2k′+r′)×(m-2k+r-2k′+r′) is arbitrary.


## 5. The Skew-Symmetric Orthogonal Solutions of the System (1.1)

In this section, we show the skew-symmetric orthogonal solutions of the system (1.1).Theorem 5.1.
GivenA,B∈ℝn×2m,  C,D∈ℝ2m×n. Suppose the matrix equation AX=B has skew-symmetric orthogonal solutions with the form
(5.1)X=V~1(I00H)Q~1T,
where H∈SSOℝ2r×2r is arbitrary, V~1,  Q~1∈Oℝ2m×2m. Partition
(5.2)Q~1TC=(Q1Q2),V~1TD=(V1V2),
where Q1,V1∈ℝ(2m-2r)×n,  Q2,V2∈ℝ2r×n. Then the system (1.1) has skew-symmetric orthogonal solutions if and only if
(5.3)AAT=BBT,ABT=-BAT,Q1=V1,Q2TQ2=V2TV2,Q2TV2=-V2TQ2.
In which case, the solutions can be expressed as
(5.4)X=V^1(I00J′)Q^1T,
where
(5.5)V^=V~1(I2m-2r00W~)∈Oℝ2m×2m,Q^=Q~1(I2m-2r00U~)∈Oℝ2m×2m,
and J′∈SSOℝ2k′×2k′ is arbitrary.Proof.
By [2, Theorem 2], the matrix equation AX=B has the skew-symmetric orthogonal solutions if and only if AAT=BBT and ABT=-BAT. When these conditions are satisfied, the general skew-symmetric orthogonal solutions can be expressed as (5.1). Substituting (5.1) and (5.2) into the matrix equation XC=D, we get Q1=V1 and HQ2=V2. From Lemma 2.7, equation HQ2=V2 has a skew-symmetric orthogonal solution H if and only if
(5.6)Q2TQ2=V2TV2,Q2TV2=-V2TQ2.
When these conditions are satisfied, the solution can be described as
(5.7)H=W~(I00J′)U~T,
where J′∈SSOℝ2k′×2k′is arbitrary, W~,  U~∈Oℝ2r×2r. Inserting (5.7) into (5.1) yields that the system (1.1) has skew-symmetric orthogonal solutions if and only if all equalities in (5.3) hold. In which case, the solutions can be expressed as (5.4).Similarly, the following theorem holds.Theorem 5.2.
GivenA,B∈ℝn×2m,  C,D∈ℝ2m×n. Suppose the matrix equation XC=D has skew-symmetric orthogonal solutions with the form
(5.8)X=W~(I00K)U~T,
where K∈SSOℝ2p×2p is arbitrary, W~,  U~∈Oℝ2m×2m. Partition
(5.9)AW~=(W1W2),BU~=(U1U2),
where W1,U1∈ℝn×(2m-2p),  W2,U2∈ℝn×2p. Then the system (1.1) has skew-symmetric orthogonal solutions if and only if
(5.10)DTD=CTC,DTC=-CTD,W1=U1,W2W2T=U2U2T,U2W2T=-W2U2T.
In which case, the solutions can be expressed as
(5.11)X=W^1(I00J′′)U^1T,
where
(5.12)W^1=W~(I2m-2p00W~1)∈Oℝ2m×2m,U^1=(I2m-2p00U~1T)U~∈Oℝ2m×2m,
and J′′∈SSOℝ2q×2q is arbitrary.


## 6. The Least Squares (Skew-) Symmetric Orthogonal Solutions of the System (1.1)

If the solvability conditions of a system of matrix equations are not satisfied, it is natural to consider its least squares solution. In this section, we get the least squares (skew-) symmetric orthogonal solutions of the system (1.1), that is, seek X∈SSOℝ2m×2m(SOℝn×n) such that
(6.1)minX∈SSOℝ2m×2m(SOℝn×n)∥AX-B∥2+∥XC-D∥2.
With the help of the definition of the Frobenius norm and the properties of the skew-symmetric orthogonal matrix, we get that
(6.2)∥AX-B∥2+∥XC-D∥2=∥A∥2+∥B∥2+∥C∥2+∥D∥2-2tr(XT(ATB+DCT)).
Let
(6.3)ATB+DCT=K,KT-K2=T.
Then, it follows from the skew-symmetric matrix X that
(6.4)tr(XT(ATB+DCT))=tr(XTK)=tr(-XK)=tr(X(KT-K2))=tr(XT).
Therefore, (6.1) holds if and only if (6.4) reaches its maximum. Now, we pay our attention to find the maximum value of (6.4). Assume the eigenvalue decomposition of T is
(6.5)T=E(Λ000)ET
with
(6.6)Λ=diag(Λ1,…,Λl),Λi=(0αi-αi0),αi>0,i=1,…,l;2l=rank(T).
Denote
(6.7)ETXE=(X11X12X21X22),
partitioned according to
(6.8)(Λ000),
then (6.4) has the following form:
(6.9)tr(XT)=tr(ETXE(Λ000))=tr(X11Λ).
Thus, by
(6.10)X11=I~=diag(I~1,…,I~l),
where
(6.11)I~i=(0-110),i=1,…,l.
Equation (6.9) gets its maximum. Since ETXE is skew-symmetric, it follows from
(6.12)X=E(I~00G)ET,
where G∈SSOℝ(2m-2l)×(2m-2l) is arbitrary, that (6.1) obtains its minimum. Hence we have the following theorem.Theorem 6.1.
GivenA,B∈ℝn×2m and C,D∈ℝ2m×n, denote
(6.13)ATB+DCT=K,KT-K2=T,
and let the spectral decomposition of T be (6.5). Then the least squares skew-symmetric orthogonal solutions of the system (1.1) can be expressed as (6.12).IfX in (6.1) is a symmetric orthogonal matrix, then by the definition of the Frobenius norm and the properties of the symmetric orthogonal matrix, (6.2) holds. Let
(6.14)ATB+DCT=H,HT+H2=N.
Then we get that
(6.15)minX∈SOℝn×n∥AX-B∥2+∥XC-D∥2=minX∈SOℝn×n[∥A∥2+∥B∥2+∥C∥2+∥D∥2-2tr(XN)].
Thus (6.15) reaches its minimum if and only if tr(XN) obtains its maximum. Now, we focus on finding the maximum value of tr(XN). Let the spectral decomposition of the symmetric matrix N be
(6.16)N=M(Σ000)MT,
where
(6.17)Σ=(Σ+00Σ-),Σ+=diag(λ1,…,λs),λ1≥λ2≥⋯≥λs>0,Σ-=diag(λs+1,…,λt),λt≤λt-1≤⋯≤λs+1<0,t=rank(N).
Denote
(6.18)MTXM=(X11¯X12¯X21¯X22¯)
being compatible with
(6.19)(Σ000).
Then
(6.20)tr(XN)=tr(MTXM(Σ000))=tr((X11¯X12¯X21¯X22¯)(Σ000))=tr(X11¯Σ).
Therefore, it follows from
(6.21)X11¯=I^=(Is00-It-s)
that (6.20) reaches its maximum. Since MTXM is a symmetric orthogonal matrix, then when X has the form
(6.22)X=M(I^00L)MT,
where L∈SOℝ(n-t)×(n-t) is arbitrary, (6.15) gets its minimum. Thus we obtain the following theorem.Theorem 6.2.
GivenA,B∈ℝn×m,  C,D∈ℝm×n, denote
(6.23)ATB+DCT=H,HT+H2=N,
and let the eigenvalue decomposition of N be (6.16). Then the least squares symmetric orthogonal solutions of the system (1.1) can be described as (6.22).Algorithm 6.3.
Consider the following.Step1.
InputA,B∈ℝn×m and C,D∈ℝm×n.Step2.
Compute(6.24)ATB+DCT=H,N=HT+H2.Step3.
Compute the spectral decomposition ofN with the form (6.16).Step4.
Compute the least squares symmetric orthogonal solutions of (1.1) according to (6.22).Example 6.4.
Assume(6.25)A=(12.28.4-5.66.39.410.711.82.98.56.99.6-7.810.62.311.57.86.78.93.67.84.911.99.45.94.56.77.83.15.611.6),B=(8.59.43.67.86.34.72.73.67.99.45.67.83.76.78.69.83.42.9-4.36.25.77.45.49.52.93.9-5.26.37.84.6),C=(5.46.43.75.69.73.64.27.8-6.37.86.73.5-4.62.92.8-2.77.210.83.73.81.93.98.25.611.28.97.89.47.95.6),D=(7.99.55.42.88.68.72.66.78.48.15.73.9-2.95.21.94.85.81.8-7.25.82.87.94.56.79.69.54.13.49.83.9).It can be verified that the given matricesA,B,C, and D do not satisfy the solvability conditions in Theorem 4.1 or Theorem 4.2. So we intend to derive the least squares symmetric orthogonal solutions of the system (1.1). By Algorithm 6.3, we have the following results:(1)
the least squares symmetric orthogonal solution(6.26)X=(0.012000.06621-0.249780.270470.91302-0.162180.066210.656010.22702-0.557690.245870.37715-0.249780.227020.806610.402540.04066-0.267840.27047-0.557690.402540.068530.237990.626450.913020.245870.040660.23799-0.12142-0.18135-0.162180.37715-0.267840.62645-0.1813540.57825),(2)
(6.27)minX∈SOℝm×m∥AX-B∥2+∥XC-D∥2=1.98366,∥XTX-I6∥=2.84882×10-15,∥XT-X∥=0.00000.Remark 6.5.
(1) There exists a unique symmetric orthogonal solution such that (6.1) holds if and only if the matrix
(6.28)N=HT+H2,
where
(6.29)H=ATB+DCT,
is invertible. Example 6.4 just illustrates it.
(2) The algorithm about computing the least squares skew-symmetric orthogonal solutions of the system (1.1) can be shown similarly; we omit it here.


## 7. Conclusions

This paper is devoted to giving the solvability conditions of and the expressions of the orthogonal solutions, the symmetric orthogonal solutions, and the skew-symmetric orthogonal solutions to the system (1.1), respectively, and meanwhile obtaining the least squares symmetric orthogonal and skew-symmetric orthogonal solutions of the system (1.1). In addition, an algorithm and an example have been provided to compute its least squares symmetric orthogonal solutions.



---

*Source: 471573-2012-12-24.xml*