# Solving Optimization Problems on Hermitian Matrix Functions with Applications

Authors: Xiang Zhang; Shu-Wen Xiang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We consider the extremal inertias and ranks of the matrix expressionsf(X,Y)=A3-B3X-(B3X)*-C3YD3-(C3YD3)*, where A3=A3*,B3,C3, and D3 are known matrices and Y and X are the solutions to the matrix equations A1Y=C1, YB1=D1, and A2X=C2, respectively. As applications, we present necessary and sufficient condition for the previous matrix function f(X,Y) to be positive (negative), non-negative (positive) definite or nonsingular. We also characterize the relations between the Hermitian part of the solutions of the above-mentioned matrix equations. Furthermore, we establish necessary and sufficient conditions for the solvability of the system of matrix equations A1Y=C1, YB1=D1, A2X=C2, and B3X+(B3X)*+C3YD3+(C3YD3)*=A3, and give an expression of the general solution to the above-mentioned system when it is solvable.


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## Body

## 1. Introduction

Throughout, we denote the field of complex numbers byℂ, the set of all m×n matrices over ℂ by ℂm×n, and the set of all m×m Hermitian matrices by ℂhm×m. The symbols A* and ℛ(A) stand for the conjugate transpose, the column space of a complex matrix A respectively. In denotes the n×n identity matrix. The Moore-Penrose inverse [1] A† of A, is the unique solution X to the four matrix equations:
(1)(i)AXA=A,(ii)XAX=X,(iii)(AX)*=AX,(iv)(XA)*=XA.
Moreover, LA and RA stand for the projectors LA=I-A†A,RA=I-AA† induced by A. It is well known that the eigenvalues of a Hermitian matrix A∈ℂn×n are real, and the inertia of A is defined to be the triplet
(2)𝕀n(A)={i+(A),i-(A),i0(A)},
where i+(A), i-(A), and i0(A) stand for the numbers of positive, negative, and zero eigenvalues of A, respectively. The symbols i+(A) and i-(A) are called the positive index and the negative index of inertia, respectively. For two Hermitian matrices A and B of the same sizes, we say A≥B (A≤B) in the Löwner partial ordering if A-B is positive (negative) semidefinite. The Hermitian part of X is defined as H(X)=X+X*. We will say that X is Re-nnd (Re-nonnegative semidefinite) if H(X)≥0,X is Re-pd (Re-positive definite) if H(X)>0, and X is Re-ns if H(X) is nonsingular.It is well known that investigation on the solvability conditions and the general solution to linear matrix equations is very active (e.g., [2–9]). In 1999, Braden [10] gave the general solution to
(3)BX+(BX)*=A.
In 2007, Djordjević [11] considered the explicit solution to (3) for linear bounded operators on Hilbert spaces. Moreover, Cao [12] investigated the general explicit solution to
(4)BXC+(BXC)*=A.
Xu et al. [13] obtained the general expression of the solution of operator equation (4). In 2012, Wang and He [14] studied some necessary and sufficient conditions for the consistence of the matrix equation
(5)A1X+(A1X)*+B1YC1+(B1YC1)*=E1
and presented an expression of the general solution to (5).Note that (5) is a special case of the following system:
(6)A1Y=C1,YB1=D1,A2X=C2,B3X+(B3X)*+C3YD3+(C3YD3)*=A3.
To our knowledge, there has been little information about (6). One goal of this paper is to give some necessary and sufficient conditions for the solvability of the system of matrix (6) and present an expression of the general solution to system (6) when it is solvable.In order to find necessary and sufficient conditions for the solvability of the system of matrix equations (6), we need to consider the extremal ranks and inertias of (10) subject to (13) and (11).There have been many papers to discuss the extremal ranks and inertias of the following Hermitian expressions:(7)p(X)=A3-B3X-(B3X)*,(8)g(Y)=A-BYC-(BYC)*,(9)h(X,Y)=A1-B1XB1*-C1YC1*,(10)f(X,Y)=A3-B3X-(B3X)*-C3YD3-(C3YD3)*.
Tian has contributed much in this field. One of his works [15] considered the extremal ranks and inertias of (7). He and Wang [16] derived the extremal ranks and inertias of (7) subject to A1X=C1, A2XB2=C2. Liu and Tian [17] studied the extremal ranks and inertias of (8). Chu et al. [18] and Liu and Tian [19] derived the extremal ranks and inertias of (9). Zhang et al. [20] presented the extremal ranks and inertias of (9), where X and Y are Hermitian solutions of
(11)A2X=C2,(12)YB2=D2,
respectively. He and Wang [16] derived the extremal ranks and inertias of (10). We consider the extremal ranks and inertias of (10) subject to (11) and
(13)A1Y=C1,YB1=D1,
which is not only the generalization of the above matrix functions, but also can be used to investigate the solvability conditions for the existence of the general solution to the system (6). Moreover, it can be applied to characterize the relations between Hermitian part of the solutions of (11) and (13).The remainder of this paper is organized as follows. In Section2, we consider the extremal ranks and inertias of (10) subject to (11) and (13). In Section 3, we characterize the relations between the Hermitian part of the solution to (11) and (13). In Section 4, we establish the solvability conditions for the existence of a solution to (6) and obtain an expression of the general solution to (6).


## 2. Extremal Ranks and Inertias of Hermitian Matrix Function (10) with Some Restrictions

In this section, we consider formulas for the extremal ranks and inertias of (10) subject to (11) and (13). We begin with the following Lemmas.Lemma 1 (see [21]).
(a) LetA1, C1, B1, and D1 be given. Then the following statements are equivalent:(1)
system (13) is consistent,(2)
(14)RA1C1=0,D1LB1=0,A1D1=C1B1.(3)
(15)r[A1C1]=r(A1),[D1B1]=r(B1),A1D1=C1B1.In this case, the general solution can be written as
(16)Y=A1†C1+LA1D1B1†+LA1VRB1,
where V is arbitrary.
(b) LetA2 and C2 be given. Then the following statements are equivalent:(1)
equation (11) is consistent,(2)
(17)RA2C2=0,(3)
(18)r[A2C2]=r(A2).In this case, the general solution can be written as
(19)X=A†C+LAW,
where W is arbitrary.Lemma 2 ([22, Lemma 1.5, Theorem 2.3]).
LetA∈ℂhm×m, B∈ℂm×n, and D∈ℂhn×n, and denote that
(20)M=[ABB*0],N=[0QQ*0],L=[ABB*D],G=[PMLNLNM*0].
Then one has the following(a)
the following equalities hold(21)i±(M)=r(B)+i±(RBARB),(22)i±(N)=r(Q),(b)
ifℛ(B)⊆ℛ(A), then i±(L)=i±(A)+i±(D-B*A†B)). Thus i±(L)=i±(A) if and only if ℛ(B)⊆ℛ(A) and i±(D-B*A†B)=0,(c)
(23)i±(G)=[PM0M*0N*0N0]-r(N).Lemma 3 (see [23]).
LetA∈ℂm×n, B∈ℂm×k, and C∈ℂl×n. Then they satisfy the following rank equalities:(a)
r[AB]=r(A)+r(EAB)=r(B)+r(EBA),(b)
r[AC]=r(A)+r(CFA)=r(C)+r(AFC),(c)
r[ABC0]=r(B)+r(C)+r(EBAFC),(d)
r[BAFC]=r[BA0C]-r(C),(e)
r[CEBA]=r[C0AB]-r(B),(f)
r[ABFDEEC0]=r[AB0C0E0D0]-r(D)-r(E),Lemma 4 (see [15]).
LetA∈ℂhm×m, B∈ℂm×n, C∈ℂhn×n, Q∈ℂm×n, and P∈ℂp×n be given, and T∈ℂm×m be nonsingular. Then one has the following (1)
i±(TAT*)=i±(A),(2)
i±[A00C]=i±(A)+i±(C),(3)
i±[0QQ*0]=r(Q),(4)
i±[ABLPLPB*0]+r(P)=i±[AB0B*0P*0P0].Lemma 5 (see [22, Lemma  1.4]).
LetS be a set consisting of (square) matrices over ℂm×m, and let H be a set consisting of (square) matrices over ℂhm×m. Then Then one has the following (a)
S has a nonsingular matrix if and only if maxX∈Sr(X)=m;(b)
anyX∈S is nonsingular if and only if minX∈Sr(X)=m;(c)
{0}∈S if and only if minX∈Sr(X)=0;(d)
S={0} if and only if maxX∈Sr(X)=0;(e)
H has a matrix X>0(X<0) if and only if maxX∈Hi+(X)=m(maxX∈Hi-(X)=m);(f)
anyX∈H satisfies X>0(X<0) if and only if minX∈Hi+(X)=m(minX∈Hi-(X)=m);(g)
H has a matrix X≥0(X≤0) if and only if minX∈Hi-(X)=0(minX∈Hi+(X)=0);(h)
anyX∈H satisfies X≥0(X≤0) if and only if maxX∈Hi-(X)=0(maxX∈Hi+(X)=0).Lemma 6 (see [16]).
Letp(X,Y)=A-BX-(BX)*-CYD-(CYD)*, where A, B, C, and D are given with appropriate sizes, and denote that
(24)M1=[ABCB*00C*00],M2=[ABD*B*00D00],M3=[ABCD*B*000],M4=[ABCD*B*000C*000],M5=[ABCD*B*000D000].
Then one has the following:(1)
the maximal rank ofp(X,Y) is
(25)maxX∈ℂn×m,Yr[p(X,Y)]=min{m,r(M1),r(M2),r(M3)},(2)
the minimal rank ofp(X,Y) is
(26)minX∈ℂn×m,Yr[p(X,Y)]=2r(M3)-2r(B)+max{u++u-,v++v-,u++v-,u-+v+},(3)
the maximal inertia ofp(X,Y) is
(27)maxX∈ℂn×m,Yi±[p(X,Y)]=min{i±(M1),i±(M2)},(4)
the minimal inertias ofp(X,Y) is
(28)minX∈ℂn×m,Yi±[p(X,Y)]=r(M3)-r(B)+max{i±(M1)-r(M4),i±(M2)-r(M5)},where
(29)u±=i±(M1)-r(M4),v±=i±(M2)-r(M5).Now we present the main theorem of this section.Theorem 7.
LetA1∈ℂm×n, C1∈ℂm×k, B1∈ℂk×l, D1∈ℂn×l, A2∈ℂt×q, C2∈ℂt×p, A3∈ℂhp×p, B3∈ℂp×q,   C3∈ℂp×n, and D3∈ℂp×n be given, and suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by S and (11) by G. Put
(30)E1=[A3C3D3*C1*B3C2*C3*0A1*00C1D3A1000B3*000A2*C200A20],E2=r[A3D3*C3D1B3C2*D30B100D1*C3*B1*000B3*000A2*C200A20],E3=[A3B3C3*D3*C2*B3*000A2*D1*C3*00B1*0C1D30A100C2A2000],E4=[A3B3C3D3*C2*D3*C1*B3*000A2*0C3*0000A1*000B1*00C1D30A1000C2A20000],E5=[A3B3C3D3*C2*C3D1B3*000A2*0D30000B1D1*C3*00A1*0000A1000C2A20000].
Then one has the following:(a)
the maximal rank of (10) subject to (13) and (11) is
(31)maxX∈G,Y∈Sr[f(X,Y)]=min{p,r(E1)-2r(A1)-2r(A2),r(E2)-2r(B1)-2r(A2),r(E3)-2r(A2)-r(A1)-r(B1)},(b)
the minimal rank of (10) subject to (13) and (11) is
(32)minX∈G,Y∈Sr[f(X,Y)]=2r(E3)-2r[B3A2]+max{r(E1)-2r(E4),r(E2)-2r(E5),i+(E1)+i-(E2)-r(E4)-r(E5),i-(E1)+i+(E2)-r(E4)-r(E5)},(c)
the maximal inertia of (10) subject to (13) and (11) is
(33)maxX∈G,Y∈Sxi±[f(X,Y)]=min{i±(E1)-r(A1)-r(A2),i±(E2)-r(B1)-r(A2)},(d)
the minimal inertia of (10) subject to (13) and (11) is
(34)minX∈G,Y∈Si±[f(X,Y)]=r(E3)-r[B3A2]+max{i±(E1)-r(E4),i±(E2)-r(E5)}.Proof.
By Lemma1, the general solutions to (13) and (11) can be written as
(35)X=A2†C2+LA2W,Y=A1†C1+LA1D1B1†+LA1ZRB1,
where W and Z are arbitrary matrices with appropriate sizes. Put
(36)Q=B3LA2,T=C3LA1,J=RB1D3,P=A3-B3A2†C2-(B3A2†C2)*-C3(A1†C1+LA1D1B1†)D3-(C3(A1†C1+LA1D1B1†)D3)*.
Substituting (36) into (10) yields
(37)f(X,Y)=P-QW-(QW)*-TZJ-(TZJ)*.
ClearlyP is Hermitian. It follows from Lemma 6 that
(38)maxX∈G,Y∈Sr[f(X,Y)]=maxW,Zr(P-QW-(QW)*-TZJ-(TZJ)*)=min{m,r(N1),r(N2),r(N3)},(39)minX∈G,Y∈Sr[f(X,Y)]=maxW,Zr(P-QW-(QW)*-TZJ-(TZJ)*)=2r(N3)-2r(Q)+max{s++s-,t++t-,s++t-,s-+t+},(40)maxX∈G,Y∈Si±[f(X,Y)]=maxW,Zr(P-QW-(QW)*-TZJ-(TZJ)*)=min{i±(N1),i±(N2)},(41)minX∈G,Y∈Si±[f(X,Y)]=maxW,Zr(P-QW-(QW)*-TZJ-(TZJ)*)=r(N3)-r(Q)+max{s±,t±},
where
(42)N1=[PQTQ*00T*00],N2=[PQJ*Q*00J00],N3=[PQTJ*Q*000],N4=[PQTJ*Q*000T*000],N5=[PQTJ*Q*000J000],s±=i±(N1)-r(N4),t±=i±(N2)-r(N5).
Now, we simplify the ranks and inertias of block matrices in (38)–(41).
By Lemma4, block Gaussian elimination, and noting that
(43)LS*=(I-S†S)*=I-S*(S*)†=RS*,
we have the following:
(44)r(N1)=r[PQTQ*00T*00]=r[A3C3D3*C1*B3C2*C3*0A1*00C1D3A1000B3*000A2*C200A20]-2r(A1)-2r(A2).
By C1B1=A1D1, we obtain
(45)r(N2)=r[PQJ*Q*00J00]=r[A3D3*C3D1B3C2*D30B100D1*C3*B1*000B3*000A2*C200A20]-2r(B1)-2r(A2),r(N3)=r[PQTJ*Q*000]=r[A3B3C3*D3*C2*B3*000A2*D1*C3*00B1*0C1D30A100C2A2000]-r(B1)-2r(A2)-r(A1),r(N4)=r[PQTJ*Q*000T*000]=r[A3B3C3D3*C2*D3*C1*B3*000A2*0C3*0000A1*000B1*00C1D30A1000C2A20000]-r(B1)-2r(A2)-2r(A1),r(N5)=r[PQTJ*Q*000T*000]=r[A3B3C3D3*C2*C3D1B3*000A2*0D30000B1D1*C3*00A1*0000A1000C2A20000]-2r(B1)-2r(A2)-r(A1).
By Lemma 2, we can get the following:
(46)i±(N1)=i±[PQTQ*00T*00]=i±[A3C3D3*C1*B3C2*C3*0A1*00C1D3A1000B3*000A2*C200A20]-r(A1)-r(A2),(47)i±(N2)=i±[PQJ*Q*00J00]=i±[A3D3*C3D1B3C2*D30B100D1*C3*B1*000B3*000A2*C200A20]-r(B1)-r(A2).
Substituting (44)-(47) into (38) and (41) yields (31)–(34), respectively.Corollary 8.
LetA1, C1, B1, D1, A2, C2, A3, B3, C3, D3, and Ei, (i=1,2,…,5) be as in Theorem 7, and suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by S and (11) by G. Then, one has the following:(a)
there existX∈G and Y∈S such that A3-B3X-(B3X)*-C3YD3-(C3YD3)*>0 if and only if
(48)i+(E1)-r(A1)-r(A2)≥p,i+(E2)-r(B1)-r(A2)≥p.(b)
there existX∈G and Y∈S such that A3-B3X-(B3X)*-C3YD3-(C3YD3)*<0 if and only if
(49)i-(E1)-r(A1)-r(A2)≥p,i-(E2)-r(B1)-r(A2)≥p,(c)
there existX∈G and Y∈S such that A3-B3X-(B3X)*-C3YD3-(C3YD3)*≥0 if and only if
(50)r(E3)-r[B3A2]+i-(E1)-r(E4)≤0,r(E3)-r[B3A2]+i-(E2)-r(E5)≤0.(d)
there existX∈G and Y∈S such that A3-B3X-(B3X)*-C3YD3-(C3YD3)*≤0 if and only if
(51)r(E3)-r[B3A2]+i+(E1)-r(E4)≤0,r(E3)-r[B3A2]+i+(E2)-r(E5)≤0,(e)
A3-B3X-(B3X)*-C3YD3-(C3YD3)*>0 for all X∈G and Y∈S if and only if
(52)r(E3)-r[B3A2]+i+(E1)-r(E4)=porr(E3)-r[B3A2]+i+(E2)-r(E5)=p,(f)
A3-B3X-(B3X)*-C3YD3-(C3YD3)*<0 for all X∈G and Y∈S if and only if
(53)r(E3)-r[B3A2]+i-(E1)-r(E4)=porr(E3)-r[B3A2]+i-(E2)-r(E5)=p,(g)
A3-B3X-(B3X)*-C3YD3-(C3YD3)*≥0 for all X∈G and Y∈S if and only if
(54)i-(E1)-r(A1)-r(A2)≤0ori-(E2)-r(B1)-r(A2)≤0,(h)
A3-B3X-(B3X)*-C3YD3-(C3YD3)*≤0 for all X∈G and Y∈S if and only if
(55)i+(E1)-r(A1)-r(A2)≤0ori+(E2)-r(B1)-r(A2)≤0,(i)
there existX∈G and Y∈S such that A3-B3X-(B3X)*-C3YD3-(C3YD3)* is nonsingular if and only if
(56)r(E1)-2r(A1)-2r(A2)≥p,r(E2)-2r(B1)-2r(A2)≥p,r(E3)-2r(A2)-r(A1)-r(B1)≥p.


## 3. Relations between the Hermitian Part of the Solutions to (13) and (11)

Now we consider the extremal ranks and inertias of the difference between the Hermitian part of the solutions to (13) and (11).Theorem 9.
LetA1∈ℂm×p, C1∈ℂm×p, B1∈ℂp×l, D1∈ℂp×l, A2∈ℂt×p, and C2∈ℂt×p, be given. Suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by S and (11) by G. Put
(57)H1=[0IC1*-IC2*I0A1*00C1A1000-I000A2*C200A20],H2=r[0ID1-IC2*I0B100D1*B1*000-I000A2*C200A20],H3=[0-IIIC2*-I000A2*D1*00B1*0C10A100C2A2000],H4=[0-IIIC2*C1*-I000A2*0I0000A1*000B1*00C10A1000C2A20000],H5=[0-IIIC2*D1-I000A2*0I0000B1D1*00A1*0000A1000C2A20000].
Then one has the following:
(58)maxX∈G,Y∈Sr[(X+X*)-(Y+Y*)]=min{p,r(H1)-2r(A1)-2r(A2),r(H2)-2r(B1)-2r(A2),r(H3)-2r(A2)-r(A1)-r(B1)},minX∈G,Y∈Sr[(X+X*)-(Y+Y*)]=2r(H3)-2p+max{r(H1)-2r(H4),r(H2)-2r(H5),i+(H1)+i-(H2)-r(H4)-r(H5),i-(H1)+i+(H2)-r(H4)-r(H5)},maxX∈G,Y∈Si±[(X+X*)-(Y+Y*)]=min{i±(H1)-r(A1)-r(A2),i±(H2)-r(B1)-r(A2)},minX∈G,Y∈Si±[(X+X*)-(Y+Y*)]=r(E3)-p+max{i±(H1)-r(H4),i±(H2)-r(H5)}.Proof.
By lettingA3=0, B3=-I, C3=I, and D3=I in Theorem 7, we can get the results.Corollary 10.
LetA1, C1, B1, D1, A2, C2, and Hi, (i=1,2,…,5) be as in Theorem 9, and suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by S and (11) by G. Then, one has the following:(a)
there existX∈G and Y∈S such that (X+X*)>(Y+Y*) if and only if
(59)i+(H1)-r(A1)-r(A2)≥p,i+(H2)-r(B1)-r(A2)≥p.(b)
there existX∈G and Y∈S such that (X+X*)<(Y+Y*) if and only if
(60)i-(H1)-r(A1)-r(A2)≥p,i-(H2)-r(B1)-r(A2)≥p,(c)
there existX∈G and Y∈S such that (X+X*)≥(Y+Y*) if and only if
(61)r(H3)-p+i-(H1)-r(H4)≤0,r(H3)-p+i-(H1)-r(H4)≤0,(d)
there existX∈G and Y∈S such that (X+X*)≤(Y+Y*) if and only if
(62)r(H3)-p+i+(H1)-r(H4)≤0,r(H3)-p+i+(H2)-r(H5)≤0,(e)
(X+X*)>(Y+Y*) for all X∈G and Y∈S if and only if
(63)r(H3)-p+i+(H1)-r(H4)=porr(H3)-p+i+(H2)-r(H5)=p,(f)
(X+X*)<(Y+Y*) for all X∈G and Y∈S if and only if
(64)r(H3)-p+i-(H1)-r(H4)=porr(H3)-p+i-(H2)-r(H5)=p,(g)
(X+X*)≥(Y+Y*) for all X∈G and Y∈S if and only if
(65)i-(H1)-r(A1)-r(A2)≤0ori-(H2)-r(B1)-r(A2)≤0,(h)
(X+X*)≤(Y+Y*) for all X∈G and Y∈S if and only if
(66)i+(H1)-r(A1)-r(A2)≤0ori+(H2)-r(B1)-r(A2)≤0,(i)
there existX∈G and Y∈S such that (X+X*)-(Y+Y*) is nonsingular if and only if
(67)r(H1)-2r(A1)-2r(A2)≥p,r(H2)-2r(B1)-2r(A2)≥p,r(H3)-2r(A2)-r(A1)-r(B1)≥p.


## 4. The Solvability Conditions and the General Solution to System (6)

We now turn our attention to (6). We in this section use Theorem 9 to give some necessary and sufficient conditions for the existence of a solution to (6) and present an expression of the general solution to (6). We begin with a lemma which is used in the latter part of this section.Lemma 11 (see [14]).
LetA1∈ℂm×n1, B1∈ℂm×n2, C1∈ℂq×m, and E1∈ℂhm×m be given. Let A=RA1B1, B=C1RA1, E=RA1E1RA1, M=RAB*, N=A*LB, and S=B*LM. Then the following statements are equivalent:(1)
equation (5) is consistent,(2)
(68)RMRAE=0,RAERA=0,LBELB=0,(3)
(69)r[E1B1C1*A1A1*000]=r[B1C1*A1]+r(A1),r[E1B1A1A1*00B1*00]=2r[B1A1],r[E1C1*A1A1*00C100]=2r[C1*A1].In this case, the general solution of (5) can be expressed as
(70)Y=12[A†EB†-A†B*M†EB†-A†S(B†)*EN†A*B†+A†E(M†)*+(N†)*EB†S†S]+LAV1+V2RB+U1LSLM+RNU2*LM-A†SU2RNA*B†,X=A1†[E1-B1YC1-(B1YC1)*]-12A1†[E1-B1YC1-(B1YC1)*]A1A1†-A1†W1A1*+W1*A1A1†+LA1W2,
where U1, U2, V1, V2, W1, and W2 are arbitrary matrices over ℂ with appropriate sizes.Now we give the main theorem of this section.Theorem 12.
LetAi, Ci, (i=1,2,3), Bj, and Dj, (j=1,3) be given. Set
(71)A=B3LA2,B=C3LA1,C=RB1D3,F=RAB,G=CRA,M=RFG*,N=F*LG,S=G*LM,(72)D=A3-B3A2†C2-(B3A2†C2)*-C3(A1†C1+LA1D1B1†)D3-D3*(A1†C1+LA1D1B1†)*C3*,(73)E=RADRA.
Then the following statements are equivalent:(1)
system (6) is consistent,(2)
the equalities in (14) and (17) hold, and
(74)RMRFE=0,RFERF=0,LGELG=0,(3)
the equalities in (15) and (18) hold, and
(75)r[A3C3D3*B3C2*B3*000A2*C1D3A1000D1*C3*0B1*00C200A20]=r[C3D3*B3A1000B1*000A2]+r[A2B3],r[A3C3B3D3*C1*C2*C3*00A1*0B3*000A2*C1D3A1000C20A200]=2r[C3B3A100A2],r[A3D3*B3C3D1C2*D300B10B3*000A2*D1*C3*B1*000C20A200]=2r[D3*B3B1*00A2].In this case, the general solution of system (6) can be expressed as
(76)X=A2†C2+LA2U,Y=A1†C1+LA1D1B1†+LA1VRB1,
where
(77)V=12[F†EG†-F†G*M†EG†-F†S(G†)*EN†F*G†+F†E(M†)*+(N†)*EG†S†S]+LFV1+V2RG+U1LSLM+RNU2*LM-F†SU2RNF*G†,U=A†[D-BVC-(BVC)*]-12A†[D-BVC-(BVC)*]AA†-A†W1A*+W1*AA†+LAW2,
where U1, U2, V1, V2, W1, and W2 are arbitrary matrices over ℂ with appropriate sizes.Proof.
(2)⇔ (3): Applying Lemma 3 and Lemma 11 gives
(78)RMRFE=0⟺r(RMRFE)=0⟺r[DBC*AA*000]=r[BC*A]+r(A)⟺r[DC3D3*B30B3*000A2*0A100000B1*00000A20]=r[C3D3*B3A1000B1*000A2]+r[A2B3]⟺r[A3C3D3*B3C2*B3*000A2*C1D3A1000D1*C3*0B1*00C200A20]=r[C3D3*B3A1000B1*000A2]+r[A2B3].
By a similar approach, we can obtain that
(79)RFERF=0⟺r[A3C3B3D3*C1*C2*C3*00A1*0B3*000A2*C1D3A1000C20A200]=2r[C3B3A100A2],LGELG=0⟺r[A3D3*B3C3D1C2*D300B10B3*000A2*D1*C3*B1*000C20A200]=2r[D3*B3B1*00A2].
(1) ⇔ (2): We separate the four equations in system (6) into three groups:
(80)A1Y=C1,YB1=D1,(81)A2X=C2,(82)B3X+(B3X)*+C3YD3+(C3YD3)*=A3.
By Lemma 1, we obtain that system (80) is solvable if and only if (14), (81) is consistent if and only if (17). The general solutions to system (80) and (81) can be expressed as (16) and (19), respectively. Substituting (16) and (19) into (82) yields
(83)AU+(AU)*+BVC+(BVC)*=D.
Hence, the system (5) is consistent if and only if (80), (81), and (83) are consistent, respectively. It follows from Lemma 11 that (83) is solvable if and only if
(84)RMRFE=0,RFERF=0,LGELG=0.
We know by Lemma 11 that the general solution of (83) can be expressed as (77).In Theorem12, let A1 and D1 vanish. Then we can obtain the general solution to the following system:
(85)A2X=C2,YB1=D1,B3X+(B3X)*+C3YD3+(C3YD3)*=A3.Corollary 13.
LetA2, C2, B1, D1, B3, C3, D3, and A3=A3* be given. Set
(86)A=B3LA2,C=RB1D3,F=RAC3,G=CRA,M=RFG*,N=F*LG,S=G*LM,D=A3-B3A2†C2-(B3A2†C2)*-C3D1B1†D3-(C3D1B1†D3)*,E=RADRA.
Then the following statements are equivalent:(1)
system (85) is consistent(2)
(87)RA2C2=0,D1LB1=0,RMRFE=0,(88)RFERF=0,LGELG=0,(3)
(89)r[A2C2]=r(A2),[D1B1]=r(B1),r[A3C3D3*B3C2*B3*000A2*D1*C3*0B1*00C200A20]=r[C3D3*B30B1*000A2]+r[A2B3],r[A3C3B3C2*C3*000B3*00A2*C20A20]=2r[C3B30A2],r[A3D3*B3C3D1C2*D300B10B3*000A2*D1*C3*B1*000C20A200]=2r[D3*B3B1*00A2].In this case, the general solution of system (6) can be expressed as
(90)X=A2†C2+LA2U,Y=D1B1†+VRB1,
where
(91)V=12[F†EG†-F†G*M†EG†-F†S(G†)*EN†F*G†+F†E(M†)*+(N†)*EG†S†S]+LFV1+V2RG+U1LSLM+RNU2*LM-F†SU2RNF*G†,U=A†[D-C3VC-(C3VC)*]-12A†[D-C3VC-(C3VC)*]AA†-A†W1A*+W1*AA†+LAW2,
where U1, U2, V1, V2, W1, and W2 are arbitrary matrices over ℂ with appropriate sizes.



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*Source: 593549-2013-04-21.xml*