Central European Journal of Mathematics

, Volume 10, Issue 1, pp 329–351 | Cite as

Inertias and ranks of some Hermitian matrix functions with applications

Research Article

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 matrix Hermitian matrix function Rank Inertia Bisymmetric solution Nonnegative definite matrix 

MSC

15A03 15A09 15A24 15B48 15B57 65F30 

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Copyright information

© © Versita Warsaw and Springer-Verlag Wien 2012

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghaiChina
  2. 2.Department of MathematicsZunyi Normal CollegeZunyiChina

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