Inertias and ranks of some Hermitian matrix functions with applications

Xiang Zhang; Qing-Wen Wang; Xin Liu

Open Mathematics (2012)

  • Volume: 10, Issue: 1, page 329-351
  • ISSN: 2391-5455

Abstract

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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 - Q X Q * - T Y T * : A X = B , Y C = 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 A X = B , Y C = D , Q X Q * + T Y T * = P to have a Hermitian solution and the system of matrix equations A X = C , B X B * = 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.

How to cite

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Xiang Zhang, Qing-Wen Wang, and Xin Liu. "Inertias and ranks of some Hermitian matrix functions with applications." Open Mathematics 10.1 (2012): 329-351. <http://eudml.org/doc/269532>.

@article{XiangZhang2012,
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 \[H = \lbrace f(X,Y) = P - QXQ* - TYT* : AX = B,YC = D,X = X*, Y = Y*\rbrace .\] 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.},
author = {Xiang Zhang, Qing-Wen Wang, Xin Liu},
journal = {Open Mathematics},
keywords = {Maximal matrix; Hermitian matrix function; Rank; Inertia; Bisymmetric solution; Nonnegative definite matrix; maximal matrix; rank; inertia; bisymmetric solution; nonnegative definite matrix; range of values; discrete Lyapunov equation; rank; Moore-Penrose pseudoinverse},
language = {eng},
number = {1},
pages = {329-351},
title = {Inertias and ranks of some Hermitian matrix functions with applications},
url = {http://eudml.org/doc/269532},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Xiang Zhang
AU - Qing-Wen Wang
AU - Xin Liu
TI - Inertias and ranks of some Hermitian matrix functions with applications
JO - Open Mathematics
PY - 2012
VL - 10
IS - 1
SP - 329
EP - 351
AB - 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 \[H = \lbrace f(X,Y) = P - QXQ* - TYT* : AX = B,YC = D,X = X*, Y = Y*\rbrace .\] 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.
LA - eng
KW - Maximal matrix; Hermitian matrix function; Rank; Inertia; Bisymmetric solution; Nonnegative definite matrix; maximal matrix; rank; inertia; bisymmetric solution; nonnegative definite matrix; range of values; discrete Lyapunov equation; rank; Moore-Penrose pseudoinverse
UR - http://eudml.org/doc/269532
ER -

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