# The Optimization on Ranks and Inertias of a Quadratic Hermitian Matrix Function and Its Applications

Authors: Yirong Yao
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We solve optimization problems on the ranks and inertias of the quadratic Hermitian matrix functionQ-XPX* subject to a consistent system of matrix equations AX=C and XB=D. As applications, we derive necessary and sufficient conditions for the solvability to the systems of matrix equations and matrix inequalities AX=C,XB=D, and XPX*=(>,<,≥,≤)Q in the Löwner partial ordering to be feasible, respectively. The findings of this paper widely extend the known results in the literature.


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## Body

## 1. Introduction

Throughout this paper, we denote the complex number field byℂ. The notations ℂm×n and ℂhm×m stand for the sets of all m×n complex matrices and all m×m complex Hermitian matrices, respectively. The identity matrix with an appropriate size is denoted by I. For a complex matrix A, the symbols A* and r(A) stand for the conjugate transpose and the rank of A, respectively. The Moore-Penrose inverse of A∈ℂm×n, denoted by A†, is defined to be the unique solution X to the following four matrix equations
(1)(1)AXA=A,(2)XAX=X,(3)(AX)*=AX,(4)(XA)*=XA.
Furthermore, LA and RA stand for the two projectors LA=I-A†A and RA=I-AA† induced by A, respectively. It is known that LA=LA* and RA=RA*. For A∈ℂhm×m, its inertia
(2)𝕀n(A)=(i+(A),i-(A),i0(A))
is the triple consisting of the numbers of the positive, negative, and zero eigenvalues of A, counted with multiplicities, respectively. It is easy to see that i+(A)+i-(A)=r(A). 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 (nonnegative) definite.The investigation on maximal and minimal ranks and inertias of linear and quadratic matrix function is active in recent years (see, e.g., [1–24]). Tian [21] considered the maximal and minimal ranks and inertias of the Hermitian quadratic matrix function
(3)h(X)=AXBX*A*+AXC+C*X*A*+D,
where B and D are Hermitian matrices. Moreover, Tian [22] investigated the maximal and minimal ranks and inertias of the quadratic Hermitian matrix function
(4)f(X)=Q-XPX*
such that AX=C.The goal of this paper is to give the maximal and minimal ranks and inertias of the matrix function (4) subject to the consistent system of matrix equations
(5)AX=C,XB=D,
where Q∈ℂhn×n,  P∈ℂhp×p are given complex matrices. As applications, we consider the necessary and sufficient conditions for the solvability to the systems of matrix equations and inequality
(6)AX=C,XB=D,XPX*=Q,AX=C,XB=D,XPX*>Q,AX=C,XB=D,XPX*<Q,AX=C,XB=D,XPX*≥Q,AX=C,XB=D,XPX*≤Q,
in the Löwner partial ordering to be feasible, respectively.


## 2. The Optimization on Ranks and Inertias of (4) Subject to (5)

In this section, we consider the maximal and minimal ranks and inertias of the quadratic Hermitian matrix function (4) subject to (5). We begin with the following lemmas.Lemma 1 (see [3]).
LetA∈ℂhm×m,  B∈ℂm×p, and C∈ℂq×m be given and denote
(7)P1=[ABB*0],P2=[AC*C0],P3=[ABC*B*00],P4=[ABC*C00].
Then
(8)maxY∈ℂp×qr[A-BYC-(BYC)*]=min{r[ABC*],r(P1),r(P2)},minY∈ℂp×qr[A-BYC-(BYC)*]=2r[ABC*]+max{w++w-,g++g-,w++g-,w-+g+},maxY∈ℂp×qi±[A-BYC-(BYC)*]=min{i±(P1),i±(P2)},minY∈ℂp×qi±[A-BYC-(BYC)*]=r[ABC*]+max{i±(P1)-r(P3),i±(P2)-r(P4)},
where
(9)w±=i±(P1)-r(P3),g±=i±(P2)-r(P4).Lemma 2 (see [4]).
LetA∈ℂm×n,  B∈ℂm×k,  C∈ℂl×n,  D∈ℂm×p,  E∈ℂq×n,  Q∈ℂm1×k, and P∈ℂl×n1 be given. Then
(10)(1)r(A)+r(RAB)=r(B)+r(RBA)=r[AB],(2)r(A)+r(CLA)=r(C)+r(ALC)=r[AC],(3)r(B)+r(C)+r(RBALC)=r[ABC0],(4)r(P)+r(Q)+r[ABLQRPC0]=r[AB0C0P0Q0],(5)r[RBALCRBDELC0]+r(B)+r(C)=r[ADBE00C00].Lemma 3 (see [23]).
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
(11)(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 4.
LetA,  C,  B, and D be given. Then the following statements are equivalent.(1)
System (5) is consistent.(2)
Let(12)r[AC]=r(A),[DB]=r(B),AD=CB.In this case, the general solution can be written as
(13)X=A†C+LADB†+LAVRB,
where V is an arbitrary matrix over ℂ with appropriate size.Now we give the fundamental theorem of this paper.Theorem 5.
Letf(X) be as given in (4) and assume that AX=C and  XB=D in (5) is consistent. Then
(14)maxAX=C,XB=Dr(Q-XPX*)=min{n+r[0PPAQCP0-D*0B*]-r(A)-r(B)-r(P),2n+r(AQA*-CPC*)-2r(A),r[Q0D0-PBD*B*0]-2r(B)},minAX=C,XB=Dr(Q-XPX*)=2n+2r[0PPAQCP0-D*0B*]-2r(A)-2r(B)-r(P)+max{s++s-,t++t-,s++t-,s-+t+},maxAX=C,XB=Di±(Q-XPX*)=min{[Q0D0-PBD*B*0]n+i±(AQA*-CPC*)-r(A),i±[Q0D0-PBD*B*0]-r(B)},(15)minAX=C,XB=Di±(Q-XPX*)=n+r[0PPAQCP0-D*0B*]-r(A)-r(B)-i±(P)+max{s±,t±},
where
(16)s±=-n+r(A)-i∓(P)+i±(AQA*-CPC*)-r[CPAQA*B*D*A*],t±=-n+r(A)-i∓(P)+i±[Q0D0-PBD*B*0]-[0PBAQCP0D*B*0].Proof.
It follows from Lemma4 that the general solution of (4) can be expressed as
(17)X=X0+LAVRB,
where V is an arbitrary matrix over ℂ and X0 is a special solution of (5). Then
(18)Q-XPX*=Q-(X0+LAVRB)P(X0+LAVRB)*.
Note that
(19)r[Q-(X0+LAVRB)P(X0+LAVRB)*]=r[Q(X0+LAVRB)PP(X0+LAVRB)*P]-r(P)=r[[QX0PPX0*P]+[LA0]V[0RBP]+([LA0]V[0RBP])*]-r(P),(20)i±[Q-(X0+LAVRB)P(X0+LAVRB)*]=i±[Q(X0+LAVRB)PP(X0+LAVRB)*P]-i±(P)=i±[[QX0PPX0*P]+[LA0]V[0RBP]+([LA0]V[0RBP])*]-i±(P).
Let
(21)q(V)=[QX0PPX0*P]+[LA0]V[0RBP]+([LA0]V[0RBP])*.
Applying Lemma 1 to (19) and (20) yields
(22)maxVr[q(V)]=min{r(M),r(M1),r(M2)},minVr[q(V)]=2r(M)+max{s++s-,t++t-,s++t-,s-+t+},maxVi±[q(V)]=min{i±(M1),i±(M2)},minVi±[q(V)]=r(M)+max{s±,t±},
where
(23)M=[QX0PLA0PX0*P0PRB],M1=[QX0PLAPX0*P0LA00],M2=[QX0P0PX0*PPRB0RBP0],M3=[QX0PLA0PX0*P0PRBLA000],M4=[QX0PLA0PX0*P0PRB0RBP00],s±=i±(M1)-r(M3),t±=i±(M2)-r(M4).
Applying Lemmas 2 and 3, elementary matrix operations and congruence matrix operations, we obtain
(24)r(M)=n+r[0PPAQCP0-D*0B*]-r(A)-r(B),r(M1)=2n+r(AQA*-CPC*)-2r(A)+r(P),i±(M1)=n+i±(AQA*-CPC*)-r(A)+i±(P),r(M2)=r[Q0D0-PBD*B*0]-2r(B)+r(P),i±(M2)=i±[Q0D0-PBD*B*0]-r(B)+i±(P),r(M3)=2n+r(P)-2r(A)-r(B)+r[CPAQA*B*D*A*],r(M4)=n+r(P)+r[0PBAQCP0D*B*0]-2r(B)-r(A).
Substituting (24) into (22), we obtain the results.Using immediately Theorem5, we can easily get the following.Theorem 6.
Letf(X) be as given in (4), s± and let t± be as given in Theorem 5 and assume that AX=C and  XB=D in (5) are consistent. Then we have the following.(a)
AX=C and  XB=D have a common solution such that Q-XPX*≥0 if and only if
(25)n+r[0PPAQCP0-D*0B*]-r(A)-r(B)-i-(P)+s-≤0,n+r[0PPAQCP0-D*0B*]-r(A)-r(B)-i-(P)+t-≤0.(b)
AX=C and  XB=D have a common solution such that Q-XPX*≤0 if and only if
(26)n+r[0PPAQCP0-D*0B*]-r(A)-r(B)-i+(P)+s+≤0,n+r[0PPAQCP0-D*0B*]-r(A)-r(B)-i+(P)+t+≤0.(c)
AX=C and  XB=D have a common solution such that Q-XPX*>0 if and only if
(27)i+(AQA*-CPC*)-r(A)≥0,i+[Q0D0-PBD*B*0]-r(B)≥n.(d)
AX=C and  XB=D have a common solution such that Q-XPX*<0 if and only if
(28)i-(AQA*-CPC*)-r(A)≥0,i-[Q0D0-PBD*B*0]-r(B)≥n.(e)
All common solutions ofAX=C and  XB=D satisfy Q-XPX*≥0 if and only if
(29)n+i-(AQA*-CPC*)-r(A)=0,or,i-[Q0D0-PBD*B*0]-r(B)=0.(f)
All common solutions ofAX=C and  XB=D satisfy Q-XPX*≤0 if and only if
(30)n+i+(AQA*-CPC*)-r(A)=0,or,i+[Q0D0-PBD*B*0]-r(B)=0.(g)
All common solutions ofAX=C and  XB=D satisfy Q-XPX*>0 if and only if
(31)n+r[0PPAQCP0-D*0B*]-r(A)-r(B)-i+(P)+s+=n,
or
(32)n+r[0PPAQCP0-D*0B*]-r(A)-r(B)-i+(P)+t+=n.(h)
All common solutions ofAX=C and  XB=D satisfy Q-XPX*<0 if and only if
(33)n+r[0PPAQCP0-D*0B*]-r(A)-r(B)-i-(P)+s-=n,
or
(34)n+r[0PPAQCP0-D*0B*]-r(A)-r(B)-i-(P)+t-=n.(i)
AX=C,  XB=D, and Q=XPX* have a common solution if and only if
(35)2n+2r[0PPAQCP0-D*0B*]-2r(A)-2r(B)-r(P)+s++s-≤0,2n+2r[0PPAQCP0-D*0B*]-2r(A)-2r(B)-r(P)+t++t-≤0,2n+2r[0PPAQCP0-D*0B*]-2r(A)-2r(B)-r(P)+s++t-≤0,2n+2r[0PPAQCP0-D*0B*]-2r(A)-2r(B)-r(P)+s-+t+≤0.LetP=I in Theorem 5, we get the following corollary.Corollary 7.
LetQ∈ℂn×n,  A,  B,  C, and D be given. Assume that (5) is consistent. Denote
(36)T1=[CAQB*D*],T2=AQA*-CC*,T3=[QDD*B*B],T4=[CAQA*B*D*A*],T5=[CBAQB*BD*].
Then,
(37)maxAX=C,XB=Dr(Q-XX*)=min{n+r(T1)-r(A)-r(B),2n+r(T2)-2r(A),n+r(T3)-2r(B)},minAX=C,XB=Dr(Q-XX*)=2r(T1)+max{r(T2)-2r(T4),-n+r(T3)-2r(T5),i+(T2)+i-(T3)-r(T4)-r(T5),-n+i-(T2)+i+(T3)-r(T4)-r(T5)}maxAX=C,XB=Di+(Q-XX*)=min{n+i+(T2)-r(A),i+(T3)-r(B)},maxAX=C,XB=Di-(Q-XX*)=min{n+i-(T2)-r(A),n+i-(T3)-r(B)},minAX=C,XB=Di+(Q-XX*)=r(T1)+max{i+(T2)-r(T4),i+(T3)-n-r(T5)},minAX=C,XB=Di-(Q-XX*)=r(T1)+max{i-(T2)-r(T4),i-(T3)-r(T5)}.Remark 8.
Corollary7 is one of the results in [24].LetB and D vanish in Theorem 5, then we can obtain the maximal and minimal ranks and inertias of (4) subject to AX=C.Corollary 9.
Letf(X) be as given in (4) and assume that AX=C is consistent. Then
(38)maxAX=Cr(Q-XPX*)=min{n+r[AQCP]-r(A)-r(B),2n+r(AQA*-CPC*)-2r(A),r(Q)+r(P)}minAX=Cr(Q-XPX*)=2n+2r[AQCP]-2r(A)+max{s++s-,t++t-,s++t-,s-+t+},maxAX=Ci±(Q-XPX*)=min{n+i±(AQA*-CPC*)-r(A),i±(Q)+i∓(P)},minAX=Ci±(Q-XPX*)=n+r[AQCP]-r(A)+i∓(P)+max{s±,t±},
where
(39)s±=-n+r(A)-i∓(P)+i±(AQA*-CPC*)-r[CPAQA*],t±=-n+r(A)+i±(Q)-r(P)-[AQCP].Remark 10.
Corollary9 is one of the results in [22].



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*Source: 961568-2013-02-04.xml*