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


Yongge Tian and Shizhen Cheng

The maximal and minimal ranks of A - BXC with applications

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.

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 rank(A - BXC) + rank(BXC) = 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

Shizhen Cheng:
Department of Mathematics, Tianjin Polytechnic University, Tianjin, China 300160