# Inertias and ranks of some Hermitian matrix functions with applications

## Authors

Research Article

- First Online:

- Received:
- Accepted:

DOI: 10.2478/s11533-011-0117-9

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

- 56 Views

## Abstract

Let The corresponding expressions of the maximal matrices of to have a Hermitian solution and the system of matrix equations 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

*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$$H = \{ f(X,Y) = P - QXQ* - TYT* : AX = B,YC = D,X = X*, Y = Y*\} .$$

*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$$AX = B, YC = D, QXQ* + TYT* = P$$

$$AX = C, BXB* = D$$

*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## Copyright information

© © Versita Warsaw and Springer-Verlag Wien 2012