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New York Journal of Mathematics 9 (2003), 345-362.

The maximal and minimal ranks of A - BXC with applications

Yongge Tian and Shizhen Cheng

Published: December 7, 2003
Keywords: Block matrix; generalized inverse; linear matrix expression; maximal rank; minimal rank; range; rank equation; Schur complement; shorted matrix.
Subject: 15A03, 15A09.

Abstract:

We consider how to take $X$ such that the linear matrix expression $A - BXC$ attains its maximal and minimal ranks, respectively. As applications, we investigate the rank invariance and the range invariance of $A - BXC$ with respect to the choice of $X$. In addition, we also give the general solution of the rank equation ${\rm rank}(A - BXC) + {\rm rank}(BXC) = {\rm rank}(A)$ and then determine the minimal rank of $A - BXC$ subject to this equation.

Author information:
Yongge Tian :
Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, Canada K7L 3N6
ytian@mast.queensu.ca

Shizhen Cheng:
Department of Mathematics, Tianjin Polytechnic University, Tianjin, China 300160
csz@mail.tjpu.edu.cn