Weighted shift matrices: Unitary equivalence, reducibility and numerical ranges

Hwa Long Gau, Ming Cheng Tsai, Han Chun Wang

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19 Scopus citations


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 b1bn= a1an and, for some fixed k, 1≤k≤n, | bj|=|ak+ j| (an+ jaj) 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 a1an= b1bn 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.

Original languageEnglish
Pages (from-to)498-513
Number of pages16
JournalLinear Algebra and Its Applications
Issue number1
StatePublished - 1 Jan 2013


  • Numerical range
  • Reducibility
  • Weighted shift matrices


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