Central European Journal of Mathematics

, Volume 10, Issue 1, pp 329–351

# Inertias and ranks of some Hermitian matrix functions with applications

Research Article

DOI: 10.2478/s11533-011-0117-9

Zhang, X., Wang, QW. & Liu, X. centr.eur.j.math. (2012) 10: 329. doi:10.2478/s11533-011-0117-9

## Abstract

Let S be a given set consisting of some Hermitian matrices with the same size. We say that a matrix AS is maximal if AW is positive semidefinite for every matrix WS. In this paper, we consider the maximal and minimal inertias and ranks of the Hermitian matrix function f(X,Y) = PQXQ* − TYT*, where * means the conjugate and transpose of a matrix, P = P*, Q, T are known matrices and for X and Y Hermitian solutions to the consistent matrix equations AX =B and YC = D respectively. As applications, we derive the necessary and sufficient conditions for the existence of maximal matrices of
$$H = \{ f(X,Y) = P - QXQ* - TYT* : AX = B,YC = D,X = X*, Y = Y*\} .$$
The corresponding expressions of the maximal matrices of H are presented when the existence conditions are met. In this case, we further prove the matrix function f(X,Y)is invariant under changing the pair (X,Y). Moreover, we establish necessary and sufficient conditions for the system of matrix equations
$$AX = B, YC = D, QXQ* + TYT* = P$$
to have a Hermitian solution and the system of matrix equations
$$AX = C, BXB* = D$$
to have a bisymmetric solution. The explicit expressions of such solutions to the systems mentioned above are also provided. In addition, we discuss the range of inertias of the matrix functions P ± QXQ* ± TYT* where X and Y are a nonnegative definite pair of solutions to some consistent matrix equations. The findings of this pape extend some known results in the literature.

### Keywords

Maximal matrixHermitian matrix functionRankInertiaBisymmetric solutionNonnegative definite matrix

### MSC

15A0315A0915A2415B4815B5765F30