TY - JOUR

T1 - Power partial isometry index and ascent of a finite matrix

AU - Gau, Hwa Long

AU - Wu, Pei Yuan

N1 - Funding Information:
The two authors acknowledge the supports from the National Science Council of the Republic of China under NSC-102-2115-M-008-007 and NSC-102-2115-M-009-007 , respectively. The second author was also supported by the MOE-ATU project.

PY - 2014/10/15

Y1 - 2014/10/15

N2 - 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 kerAk 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.

AB - 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 kerAk 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.

KW - Ascent

KW - Jordan block

KW - Partial isometry

KW - Power partial isometry

KW - Power partial isometry index

UR - http://www.scopus.com/inward/record.url?scp=84904547891&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2014.07.001

DO - 10.1016/j.laa.2014.07.001

M3 - 期刊論文

AN - SCOPUS:84904547891

VL - 459

SP - 136

EP - 144

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -