Rank Numbers of Pairs of Mutually Inverse Matrices, Asplund Theorems

Abstract

In this chapter we extend the notion of rank numbers introduced in Chapter 4 to wider sets of submatrices. Lower rank numbers for a square matrix relative to the diagonal i – j are introduced as the ranks of the maximal submatrices entirely located under that diagonal, and the upper rank numbers relative to a diagonal are defined correspondingly. If the given matrix is invertible, a strong link exists between these numbers for the matrix and its inverse. In particular, the lower and upper rank numbers relative to the main diagonal are the same for a matrix with square blocks on the main diagonal and for its inverse matrix. This implies that for such a square matrix the lower and upper quasiseparable orders coincide with the ones of the inverse matrix.