Structure ranks of matrices (Q1208303)

The author and \textit{T. L. Markham} [Linear Algebra Appl. 85, 49-56 (1987; Zbl 0607.15002)] introduced the notion of the off-diagonal rank of a square matrix \(A\) as the maximum order of a nonsingular submatrix of \(A\) none of whose entries is a diagonal entry of \(A\). In particular they showed that the off-diagonal ranks of a nonsingular matrix and its inverse coincide. In this paper the author generalizes this rank and studies the cashe cases off-block-diagonal, strictly-upper-triangular,and strictly-upper- block-triangular rank, and for these he can again show that the ranks of a nonsingular matrix and its inverse coincide.