Max-Min Problems on the Ranks and Inertias of the Matrix Expressions A−BXC±(BXC)∗ with Applications

Yonghui Liu () and Yongge Tian ()
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Yonghui Liu: Shanghai Finance University
Yongge Tian: Central University of Finance and Economics

Journal of Optimization Theory and Applications, 2011, vol. 148, issue 3, No 10, 593-622

Abstract: Abstract We introduce a simultaneous decomposition for a matrix triplet (A,B,C ∗), where A=±A ∗ and (⋅)∗ denotes the conjugate transpose of a matrix, and use the simultaneous decomposition to solve some conjectures on the maximal and minimal values of the ranks of the matrix expressions A−BXC±(BXC)∗ with respect to a variable matrix X. In addition, we give some explicit formulas for the maximal and minimal values of the inertia of the matrix expression A−BXC−(BXC)∗ with respect to X. As applications, we derive the extremal ranks and inertias of the matrix expression D−CXC ∗ subject to Hermitian solutions of a consistent matrix equation AXA ∗=B, as well as the extremal ranks and inertias of the Hermitian Schur complement D−B ∗ A ∼ B with respect to a Hermitian generalized inverse A ∼ of A. Various consequences of these extremal ranks and inertias are also presented in the paper.

Keywords: Hermitian matrix; Rank; Inertia; Generalized inverse; Schur complement; Inequality (search for similar items in EconPapers)
Date: 2011
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Citations: View citations in EconPapers (6)

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DOI: 10.1007/s10957-010-9760-8

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