Completing triangular block matrices with maximal and minimal ranks
The author considers two completion problems on block matrices with maximal and minimal ranks: For \(A=(A_{ij})\) an \(n\times n\) block matrix, where \(A_{ij}\) \((n\geq i\geq j\geq 1)\) is given, and \(A_{ij}\) \((1\leq i<j \leq n)\) is a variant block entity, determine all these variant block entries such that \(A=(A_{ij})\) has maximal and minimal possible rank, respectively. By using the theory of generalized inverses of matrices, the author presents complete solutions to these two problems. As applications he also determines maximal and minimal ranks of the matrix expression \(A-B\times C\) when \(X\) is a variant triangular block matrix, then presents a necessary and sufficient condition for the matrix equation \(B\times C=A\) to have a triangular block solution.
scientific article; zbMATH DE number 1563497