Criterion of unitary similarity for upper triangular matrices in general position

TL;DR

This paper establishes a criterion for unitary similarity of certain upper triangular matrices based on Frobenius norms of polynomial functions of their principal submatrices, extending understanding of matrix similarity in complex spaces.

Contribution

It provides a new necessary and sufficient condition for unitary similarity of upper triangular matrices in general position, involving polynomial functions and Frobenius norms.

Findings

Unitary similarity can be characterized by polynomial norms of principal submatrices.

The criterion applies to matrices not decomposable into smaller blocks.

The result extends classical similarity criteria to a broader class of matrices.

Abstract

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.