Power partial isometry index and ascent of a finite matrix

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,A2,…,Aj are all partial isometries while the ascent a(A) of A is the smallest integer k≥0 for which ker⁡Ak equals ker⁡Ak+1. It was known before that, for any matrix A, either p(A)≤min⁡{a(A),n−1} or p(A)=∞. 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.