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 kerAk+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.