TY - JOUR

T1 - Structures and numerical ranges of power partial isometries

AU - Gau, Hwa Long

AU - Wu, Pei Yuan

PY - 2014/1/1

Y1 - 2014/1/1

N2 - We derive a matrix model, under unitary similarity, of an n-by-n matrix A such that A,A2,⋯,AK (k≥1) are all partial isometries, which generalizes the known fact that if A is a partial isometry, then it is unitarily similar to a matrix of the form [0B 0C] with B* B+C*C=I. Using this model, we show that if A has ascent k and A, A 2,⋯,Ak-1 are partial isometries, then the numerical range W(A) of A is a circular disc centered at the origin if and only if A is unitarily similar to a direct sum of Jordan blocks whose largest size is k. As an application, this yields that, for any Sn-matrix A, W(A) (resp., W(⊗ A)) is a circular disc centered at the origin if and only if A is unitarily similar to the Jordan block Jn. Finally, examples are given to show that, for a general matrix A, the conditions that W(A) and W(⊗ A) are circular discs at 0 are independent of each other.

AB - We derive a matrix model, under unitary similarity, of an n-by-n matrix A such that A,A2,⋯,AK (k≥1) are all partial isometries, which generalizes the known fact that if A is a partial isometry, then it is unitarily similar to a matrix of the form [0B 0C] with B* B+C*C=I. Using this model, we show that if A has ascent k and A, A 2,⋯,Ak-1 are partial isometries, then the numerical range W(A) of A is a circular disc centered at the origin if and only if A is unitarily similar to a direct sum of Jordan blocks whose largest size is k. As an application, this yields that, for any Sn-matrix A, W(A) (resp., W(⊗ A)) is a circular disc centered at the origin if and only if A is unitarily similar to the Jordan block Jn. Finally, examples are given to show that, for a general matrix A, the conditions that W(A) and W(⊗ A) are circular discs at 0 are independent of each other.

KW - Jordan block

KW - Numerical range

KW - Partial isometry

KW - Power partial isometry

KW - S -matrix

UR - http://www.scopus.com/inward/record.url?scp=84889880554&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2013.11.013

DO - 10.1016/j.laa.2013.11.013

M3 - 期刊論文

AN - SCOPUS:84889880554

VL - 440

SP - 325

EP - 341

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 1

ER -