An n-by-n (nβ₯3) weighted shift matrix A is one of the form
[{array}{cccc}0 & a_1 & & & 0 & \ddots & & & \ddots & a_{n-1} a_n & & &
0{array}], where the ajβ's, called the weights of A, are complex numbers.
Assume that all ajβ's are nonzero and B is an n-by-n weighted shift
matrix with weights b1β,...,bnβ. We show that B is unitarily equivalent to
A if and only if b1β...bnβ=a1β...anβ and, for some fixed k, 1β€kβ€n, β£bjββ£=β£ak+jββ£ (an+jββ‘ajβ) for all j. Next, we show that
A is reducible if and only if A has periodic weights, that is, for some
fixed k, 1β€kβ€βn/2β, n is divisible by k, and
β£ajββ£=β£ak+jββ£ for all 1β€jβ€nβk. Finally, we prove that A and B
have the same numerical range if and only if a1β...anβ=b1β...bnβ and
Srβ(β£a1ββ£2,...,β£anββ£2)=Srβ(β£b1ββ£2,...,β£bnββ£2) for all 1β€rβ€βn/2β, where Srβ's are the circularly symmetric functions.Comment: 27 page