Inertias and ranks of some Hermitian matrix functions with applications

• Autorzy: Xiang Zhang , Qing-Wen Wang , Xin Liu
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Maximal matrix; Hermitian matrix function; Rank; Inertia; Bisymmetric solution; Nonnegative definite matrix

Kategorie tematyczne

• 15A24: Matrix equations and identities
• 15A09: Matrix inversion, generalized inverses
• 15B57: Hermitian, skew-Hermitian, and related matrices
• 15A03: Vector spaces, linear dependence, rank
• 65F30: Other matrix algorithms
• 15B48: Positive matrices and their generalizations; cones of matrices

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 1
• Strony: 329-351

Daty

wydano

2012-02-01

online

2011-12-09

Twórcy

autor

Xiang Zhang

autor

Qing-Wen Wang

• Shanghai University

autor

Xin Liu

• Shanghai University

Bibliografia

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Identyfikatory

DOI

10.2478/s11533-011-0117-9