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Inertias and ranks of some Hermitian matrix functions with applications

  • Research Article
  • Published:
Central European Journal of Mathematics

Abstract

Let S be a given set consisting of some Hermitian matrices with the same size. We say that a matrix A ∈ S is maximal if A − W is positive semidefinite for every matrix W ∈ S. In this paper, we consider the maximal and minimal inertias and ranks of the Hermitian matrix function f(X,Y) = P − QXQ* − 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.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai University, 99 Shangda Road, Shanghai, 200444, China

    Xiang Zhang, Qing-Wen Wang & Xin Liu

  2. Department of Mathematics, Zunyi Normal College, Zunyi, 563000, China

    Xiang Zhang

Authors
  1. Xiang Zhang
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  2. Qing-Wen Wang
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  3. Xin Liu
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Correspondence to Qing-Wen Wang.

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Zhang, X., Wang, QW. & Liu, X. Inertias and ranks of some Hermitian matrix functions with applications. centr.eur.j.math. 10, 329–351 (2012). https://doi.org/10.2478/s11533-011-0117-9

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  • DOI: https://doi.org/10.2478/s11533-011-0117-9

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