Unitary Similarity of Nonderogatory Matrices

TL;DR

This paper introduces a numerically stable geometric method for verifying unitary similarity specifically for nonderogatory matrices, improving accuracy when using approximate upper triangular forms.

Contribution

It presents a new geometric approach for the unitary similarity verification of nonderogatory matrices, emphasizing stability and efficiency.

Findings

Method is stable with respect to errors in initial matrices

Approach is efficient for matrices obtained via approximate methods

Applicable to nonderogatory matrices in numerical computations

Abstract

This paper is dedicated to the problem of verification of matrices for unitary similarity. For the case of nonderogatory matrices, we have been able to present the new solution for this problem based on geometric approach. The main advantage of this approach is stability with respect to errors in the initial upper triangular matrix. Since an upper triangular form is usually obtained by approximate methods (e.g. by QR algorithm), the mentioned advantage seems even more significant and allows us to propose the numerically stable and efficient method for verification of matrices for unitary similarity.