Journal of Applied Mathematics

Volume 2013 (2013), Article ID 961568, 6 pages

http://dx.doi.org/10.1155/2013/961568

Research Article

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

Yirong Yao

Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 3 December 2012; Accepted 9 January 2013

Academic Editor: Yang Zhang

Copyright © 2013 Yirong Yao. 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.

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

Let in Theorem 5, we get the following corollary.

Corollary 7. Let ,  ,  ,  , and be given. Assume that (5) is consistent. Denote

Then,

Remark 8. Corollary 7 is one of the results in [24].

Let and vanish in Theorem 5, then we can obtain the maximal and minimal ranks and inertias of (4) subject to .

Corollary 9. Let be as given in (4) and assume that is consistent. Then

where

Remark 10. Corollary 9 is one of the results in [22].

References

D. L. Chu, Y. S. Hung, and H. J. Woerdeman, “Inertia and rank characterizations of some matrix expressions,” SIAM Journal on Matrix Analysis and Applications, vol. 31, no. 3, pp. 1187–1226, 2009.

Z. H. He and Q. W. Wang, “Solutions to optimization problems on ranks and inertias of a matrix function with applications,” Applied Mathematics and Computation, vol. 219, no. 6, pp. 2989–3001, 2012.

Y. Liu and Y. Tian, “Max-min problems on the ranks and inertias of the matrix expressions with applications,” Journal of Optimization Theory and Applications, vol. 148, no. 3, pp. 593–622, 2011.

G. Marsaglia and G. P. H. Styan, “Equalities and inequalities for ranks of matrices,” Linear and Multilinear Algebra, vol. 2, pp. 269–292, 1974/75.

X. Zhang, Q. W. Wang, and X. Liu, “Inertias and ranks of some Hermitian matrix functions with applications,” Central European Journal of Mathematics, vol. 10, no. 1, pp. 329–351, 2012.

Q. W. Wang, Y. Zhou, and Q. Zhang, “Ranks of the common solution to six quaternion matrix equations,” Acta Mathematicae Applicatae Sinica, vol. 27, no. 3, pp. 443–462, 2011.

Q. Zhang and Q. W. Wang, “The -(skew)symmetric extremal rank solutions to a system of quaternion matrix equations,” Applied Mathematics and Computation, vol. 217, no. 22, pp. 9286–9296, 2011.

D. Z. Lian, Q. W. Wang, and Y. Tang, “Extreme ranks of a partial banded block quaternion matrix expression subject to some matrix equations with applications,” Algebra Colloquium, vol. 18, no. 2, pp. 333–346, 2011.

H. S. Zhang and Q. W. Wang, “Ranks of submatrices in a general solution to a quaternion system with applications,” Bulletin of the Korean Mathematical Society, vol. 48, no. 5, pp. 969–990, 2011.

Q. W. Wang and J. Jiang, “Extreme ranks of (skew-)Hermitian solutions to a quaternion matrix equation,” Electronic Journal of Linear Algebra, vol. 20, pp. 552–573, 2010.

I. A. Khan, Q. W. Wang, and G. J. Song, “Minimal ranks of some quaternion matrix expressions with applications,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 2031–2040, 2010.

Q. Wang, S. Yu, and W. Xie, “Extreme ranks of real matrices in solution of the quaternion matrix equation with applications,” Algebra Colloquium, vol. 17, no. 2, pp. 345–360, 2010.

Q. W. Wang, H. S. Zhang, and G. J. Song, “A new solvable condition for a pair of generalized Sylvester equations,” Electronic Journal of Linear Algebra, vol. 18, pp. 289–301, 2009.

Q. W. Wang, S. W. Yu, and Q. Zhang, “The real solutions to a system of quaternion matrix equations with applications,” Communications in Algebra, vol. 37, no. 6, pp. 2060–2079, 2009.

Q. W. Wang and C. K. Li, “Ranks and the least-norm of the general solution to a system of quaternion matrix equations,” Linear Algebra and its Applications, vol. 430, no. 5-6, pp. 1626–1640, 2009.

Q. W. Wang, H. S. Zhang, and S. W. Yu, “On solutions to the quaternion matrix equation ,” Electronic Journal of Linear Algebra, vol. 17, pp. 343–358, 2008.

Q. W. Wang, G. J. Song, and X. Liu, “Maximal and minimal ranks of the common solution of some linear matrix equations over an arbitrary division ring with applications,” Algebra Colloquium, vol. 16, no. 2, pp. 293–308, 2009.

Q. W. Wang, G. J. Song, and C. Y. Lin, “Rank equalities related to the generalized inverse with applications,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 370–382, 2008.

Q. W. Wang, S. W. Yu, and C. Y. Lin, “Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications,” Applied Mathematics and Computation, vol. 195, no. 2, pp. 733–744, 2008.

Q. W. Wang, G. J. Song, and C. Y. Lin, “Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1517–1532, 2007.

Y. Tian, “Formulas for calculating the extremum ranks and inertias of a four-term quadratic matrix-valued function and their applications,” Linear Algebra and its Applications, vol. 437, no. 3, pp. 835–859, 2012.

Y. Tian, “Solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method,” Nonlinear Analysis. Theory, Methods & Applications, vol. 75, no. 2, pp. 717–734, 2012.

Y. Tian, “Maximization and minimization of the rank and inertia of the Hermitian matrix expression with applications,” Linear Algebra and its Applications, vol. 434, no. 10, pp. 2109–2139, 2011.

Q. W. Wang, X. Zhang, and Z. H. He, “On the Hermitian structures of the solution to a pair of matrix equations,” Linear and Multilinear Algebra, vol. 61, no. 1, pp. 73–90, 2013.