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 subject to a consistent system of matrix equations and . As applications, we derive necessary and sufficient conditions for the solvability to the systems of matrix equations and matrix inequalities , and in the Löwner partial ordering to be feasible, respectively. The findings of this paper widely extend the known results in the literature.

1. Introduction

Throughout this paper, we denote the complex number field by . The notations and stand for the sets of all complex matrices and all complex Hermitian matrices, respectively. The identity matrix with an appropriate size is denoted by . For a complex matrix , the symbols and stand for the conjugate transpose and the rank of , respectively. The Moore-Penrose inverse of , denoted by , is defined to be the unique solution to the following four matrix equations Furthermore, and stand for the two projectors and induced by , respectively. It is known that and . For , its inertia is the triple consisting of the numbers of the positive, negative, and zero eigenvalues of , counted with multiplicities, respectively. It is easy to see that . For two Hermitian matrices and of the same sizes, we say in the Löwner partial ordering if 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 where and are Hermitian matrices. Moreover, Tian [22] investigated the maximal and minimal ranks and inertias of the quadratic Hermitian matrix function such that .

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 where ,   are given complex matrices. As applications, we consider the necessary and sufficient conditions for the solvability to the systems of matrix equations and inequality 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 ,  , and be given and denote Then where

Lemma 2 (see [4]). Let ,  ,  ,  ,  ,  , and be given. Then

Lemma 3 (see [23]). Let ,  ,  ,  , and be given, and, be nonsingular. Then

Lemma 4. Let ,  ,  , and be given. Then the following statements are equivalent.(1)System (5) is consistent.(2)Let In this case, the general solution can be written as where is an arbitrary matrix over with appropriate size.

Now we give the fundamental theorem of this paper.

Theorem 5. Let be as given in (4) and assume that and   in (5) is consistent. Then where

Proof. It follows from Lemma 4 that the general solution of (4) can be expressed as where is an arbitrary matrix over and is a special solution of (5). Then Note that Let Applying Lemma 1 to (19) and (20) yields where Applying Lemmas 2 and 3, elementary matrix operations and congruence matrix operations, we obtain Substituting (24) into (22), we obtain the results.

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

Theorem 6. Let be as given in (4), and let be as given in Theorem 5 and assume that and   in (5) are consistent. Then we have the following.(a) and   have a common solution such that if and only if (b) and   have a common solution such that if and only if (c) and   have a common solution such that if and only if (d) and   have a common solution such that if and only if (e)All common solutions of and   satisfy if and only if (f)All common solutions of and   satisfy if and only if (g)All common solutions of and   satisfy if and only if or (h)All common solutions of and   satisfy if and only if or (i),  , and have a common solution if and only if