Weighted shift matrices: Unitary equivalence, reducibility and numerical ranges

An n-by-n (n≥3) weighted shift matrix A is one of the form0a10⋱⋱an-1an0,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.