, Volume 148, Issue 3, pp 593-622
Date: 23 Nov 2010

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

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Abstract

We introduce a simultaneous decomposition for a matrix triplet (A,B,C ), where AA 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 ABXC±(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 ABXC−(BXC) with respect to X. As applications, we derive the extremal ranks and inertias of the matrix expression DCXC subject to Hermitian solutions of a consistent matrix equation AXA =B, as well as the extremal ranks and inertias of the Hermitian Schur complement DB 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.

Communicated by F. Zirilli.
We wish to thank an Associate Editor and referees for helpful comments and suggestions on this paper. The research of the first author was supported by the Shanghai Municipal Natural Science Foundation (10ZR1420600); Foundation of Shanghai Municipal Education Commission (11zz182); Shanghai Talent Development Funds (078); Leading Academic Discipline Project of Shanghai Municipal Education Commission (J51601).