Power partial isometry index and ascent of a finite matrix

Abstract

We give a complete characterization of nonnegative integers j and k and a positive integer n for which there is an n-by-n matrix with its power partial isometry index equal to j and its ascent equal to k. Recall that the power partial isometry index p(A) of a matrix A is the supremum, possibly infinity, of nonnegative integers j such that I, A, A(2), . . . , A(j) are all partial isometries while the ascent a(A) of A is the smallest integer k >= 0 for which ker A(k) equals ker A(k)+1. It was known before that, for any matrix A, either p(A) <= min{a(A), n - 1} or p(A) = infinity. In this paper, we prove more precisely that there is an n-by-n matrix A such that p(A) = j and a(A) = k if and only if one of the following conditions holds: (a) j = k <= n - 1, (b) j <= k - 1 and j+k <= n - 1, or (c) j <= k - 2 and j+ k = n. This answers a question we asked in a previous paper. (C) 2014 Elsevier Inc. All rights reserved.