Completing triangular block matrices with maximal and minimal ranks

Abstract

This paper considers two completion problems on block matrices with maximal and minimal ranks: Let A=(A ij) be an n×n block matrix, where A ij (n⩾i⩾j⩾1) is given, and A ij (1⩽i<j⩽n) is a variant block entry. Then determine all these variant block entries such that A=(A ij) has maximal and minimal possible ranks, respectively. By making use of the theory of generalized inverses of matrices, we present complete solutions to these two problems. As applications, we also determine maximal and minimal ranks of the matrix expression A−BXC when X is a variant triangular block matrix, and then present a necessary and sufficient condition for the matrix equation BXC=A to have a triangular block solution.