TY - JOUR

T1 - Weighted shift matrices

T2 - Unitary equivalence, reducibility and numerical ranges

AU - Gau, Hwa Long

AU - Tsai, Ming Cheng

AU - Wang, Han Chun

N1 - Funding Information:
Research supported in part by the National Science Council of the Republic of China under project NSC 101-2115-M-008-006. Corresponding author. E-mail addresses: [email protected] (H.-L. Gau), [email protected] (M.-C. Tsai), [email protected] (H.-C. Wang).

PY - 2013/1/1

Y1 - 2013/1/1

N2 - An n-by-n (n≥3) weighted shift matrix A is one of the form0 a10an- 1 an0,where the aj's, called the weights of A, are complex numbers. Assume that all aj's are nonzero and B is an n-by-n weighted shift matrix with weights b1,..., bn. We show that B is unitarily equivalent to A if and only if b1⋯ bn= a1⋯ an and, for some fixed k, 1≤k≤n, | bj|=|ak+ j| (an+ j≡ aj) for all j. Next, we show that A is reducible if and only if {| aj|}j=1n is periodic, that is, for some fixed k, 1≤k≤⌊n/2⌋, n is divisible by k, and | aj|=|ak+ j| for all j, 1≤j≤n-k. Finally, we prove that A and B have the same numerical range if and only if a1⋯ an= b1⋯ bn and Sr(| a1| 2,...,| an| 2)= Sr(| b1| 2,...,| bn| 2) for all 1≤r≤⌊n/2⌋, where Sr's are the circularly symmetric functions.

AB - An n-by-n (n≥3) weighted shift matrix A is one of the form0 a10an- 1 an0,where the aj's, called the weights of A, are complex numbers. Assume that all aj's are nonzero and B is an n-by-n weighted shift matrix with weights b1,..., bn. We show that B is unitarily equivalent to A if and only if b1⋯ bn= a1⋯ an and, for some fixed k, 1≤k≤n, | bj|=|ak+ j| (an+ j≡ aj) for all j. Next, we show that A is reducible if and only if {| aj|}j=1n is periodic, that is, for some fixed k, 1≤k≤⌊n/2⌋, n is divisible by k, and | aj|=|ak+ j| for all j, 1≤j≤n-k. Finally, we prove that A and B have the same numerical range if and only if a1⋯ an= b1⋯ bn and Sr(| a1| 2,...,| an| 2)= Sr(| b1| 2,...,| bn| 2) for all 1≤r≤⌊n/2⌋, where Sr's are the circularly symmetric functions.

KW - Numerical range

KW - Reducibility

KW - Weighted shift matrices

UR - http://www.scopus.com/inward/record.url?scp=84869086455&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2012.08.018

DO - 10.1016/j.laa.2012.08.018

M3 - 期刊論文

AN - SCOPUS:84869086455

SN - 0024-3795

VL - 438

SP - 498

EP - 513

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 1

ER -