Criterion of unitary similarity for upper triangular matrices in general position

Douglas Farenick, Vyacheslav Futorny, Tatiana G. Gerasimova, Vladimir V. Sergeichuk, Nadya Shvai

Published 2010-10-30Version 1

Each square complex matrix is unitarily similar to an upper triangular matrix with diagonal entries in any prescribed order. Let A and B be upper triangular n-by-n matrices that (i) are not similar to direct sums of matrices of smaller sizes, or (ii) are in general position and have the same main diagonal. We prove that A and B are unitarily similar if and only if ||h(A_k)||=||h(B_k)|| for all complex polynomials h(x) and k=1, 2, . . , n, where A_k and B_k are the principal k-by-k submatrices of A and B, and ||M|| is the Frobenius norm of M.