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

Abstract

We solve optimization problems on the ranks and inertias of the quadratic Hermitian matrix function Q -XP[superscript] X *[/superscript] 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 XP[superscript] X *[/superscript] = ( > , < , ...5; , ...4; )Q in the Löwner partial ordering to be feasible, respectively. The findings of this paper widely extend the known results in the literature.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Recommended by Yang Zhang

Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 3 December 2012; Accepted 9 January 2013

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Throughout this paper, we denote the complex number field by ... . The notations ^{... m ×n} and ^{... h m ×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 ∈ ^{... m ×n} , denoted by ^{A [dagger]} , is defined to be the unique solution X to the following four matrix equations [figure omitted; refer to PDF] Furthermore, _{L A} and _{R A} stand for the two projectors _{L A} =I - ^{A [dagger]} A and _{R A} =I -A ^{A [dagger]} induced by A , respectively. It is known that _{L A} = ^{L A *} and _{R A} = ^{R A *} . For A ∈ ^{... h m ×m} , its inertia [figure omitted; refer to PDF] 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 ...5;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 [figure omitted; refer to PDF] where B and D are Hermitian matrices. Moreover, Tian [ 22] investigated the maximal and minimal ranks and inertias of the quadratic Hermitian matrix function [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] where Q ∈ ^{... h n ×n} , P ∈ ^{... h p ×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 [figure omitted; refer to PDF] 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]).

Let A ∈ ^{... h m ×m} , B ∈ ^{... m ×p} , and C ∈ ^{... q ×m} be given and denote [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Lemma 2 (see [ 4]).

Let A ∈ ^{... m ×n} , B ∈ ^{... m ×k} , C ∈ ^{... l ×n} , D ∈ ^{... m ×p} , E ∈ ^{... q ×n} , Q ∈ ^{... _{m 1} ×k} , and P ∈ ^{... l × _{n 1}} be given. Then [figure omitted; refer to PDF]

Lemma 3 (see [ 23]).

Let A ∈ ^{... h m ×m} , B ∈ ^{... m ×n} , C ∈ ^{... h n ×n} , Q ∈ ^{... m ×n} , and P ∈ ^{... p ×n} be given, and, T ∈ ^{... m ×m} be nonsingular. Then [figure omitted; refer to PDF]

Lemma 4.

Let A , C , B , and D be given. Then the following statements are equivalent.

(1) System ( 5) is consistent.

(2) Let [figure omitted; refer to PDF]

In this case, the general solution can be written as [figure omitted; refer to PDF] where V is an arbitrary matrix over ... with appropriate size.

Now we give the fundamental theorem of this paper.

Theorem 5.

Let f (X ) be as given in ( 4) and assume that AX =C and XB =D in ( 5) is consistent. Then [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

It follows from Lemma 4that the general solution of ( 4) can be expressed as [figure omitted; refer to PDF] where V is an arbitrary matrix over ... and _{X 0} is a special solution of ( 5). Then [figure omitted; refer to PDF] Note that [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] Applying Lemma 1to ( 19) and ( 20) yields [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Applying Lemmas 2and 3, elementary matrix operations and congruence matrix operations, we obtain [figure omitted; refer to PDF] Substituting ( 24) into ( 22), we obtain the results.

Using immediately Theorem 5, we can easily get the following.

Theorem 6.

Let f (X ) be as given in ( 4), _{s ±} and let _{t ±} be as given in Theorem 5and 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 -XP ^{X *} ...5;0 if and only if [figure omitted; refer to PDF]

(b) AX =C and XB =D have a common solution such that Q -XP ^{X *} ...4;0 if and only if [figure omitted; refer to PDF]

(c) AX =C and XB =D have a common solution such that Q -XP ^{X *} >0 if and only if [figure omitted; refer to PDF]

(d) AX =C and XB =D have a common solution such that Q -XP ^{X *} <0 if and only if [figure omitted; refer to PDF]

(e) All common solutions of AX =C and XB =D satisfy Q -XP ^{X *} ...5;0 if and only if [figure omitted; refer to PDF]

(f) All common solutions of AX =C and XB =D satisfy Q -XP ^{X *} ...4;0 if and only if [figure omitted; refer to PDF]

(g) All common solutions of AX =C and XB =D satisfy Q -XP ^{X *} >0 if and only if [figure omitted; refer to PDF] or [figure omitted; refer to PDF]

(h) All common solutions of AX =C and XB =D satisfy Q -XP ^{X *} <0 if and only if [figure omitted; refer to PDF] or [figure omitted; refer to PDF]

(i) AX =C , XB =D , and Q =XP ^{X *} have a common solution if and only if [figure omitted; refer to PDF]

Let P =I in Theorem 5, we get the following corollary.

Corollary 7.

Let Q ∈ ^{... n ×n} , A , B , C , and D be given. Assume that ( 5) is consistent. Denote [figure omitted; refer to PDF] Then, [figure omitted; refer to PDF]

Remark 8.

Corollary 7is one of the results in [ 24].

Let B 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.

Let f (X ) be as given in ( 4) and assume that AX =C is consistent. Then [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Remark 10.

Corollary 9is one of the results in [ 22].

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