Weighted shift matrices: Unitary equivalence, reducibility and numerical ranges

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.