A criterion for unitary similarity of upper triangular matrices in general position

Each square complex matrix is unitarily similar to an upper triangular matrix with diagonal entries in any prescribed order. Let A=[aij]A=[aij] and B=[bij]B=[bij] be upper triangular n×nn×n matrices that
• are not similar to direct sums of square matrices of smaller sizes, or
• are in general position and have the same main diagonal.We prove that A and B are unitarily similar if and only if‖h(Ak)‖=‖h(Bk)‖for allh∈C[x]andk=1,…,n,where Ak:=[aij]i,j=1k and Bk:=[bij]i,j=1k are the leading principal k×kk×k submatrices of A and B  , and ‖·‖‖·‖ is the Frobenius norm.