Simultaneous unitary equivalences

TL;DR

This paper presents an algorithm to determine the existence of a single unitary matrix that simultaneously satisfies multiple unitary similarity and congruence conditions across different matrix pairs.

Contribution

It introduces a finite-computation algorithm for solving complex simultaneous unitary equivalence problems involving multiple matrix sets.

Findings

Algorithm can decide the existence of the unitary matrix with finitely many steps.

Provides a systematic approach for complex matrix equivalence problems.

Extends previous methods to more general simultaneous conditions.

Abstract

Let A, B, C, D be given finite sets of pairs of n-by-n complex matrices. We describe an algorithm to determine, with finitely many computations, whether there is a single unitary matrix U such that each pair of matrices in A is unitarily similar via U, each pair of matrices in B is unitarily congruent via U, each pair of matrices in C is unitarily similar via \bar{U}, and each pair of matrices in D is unitarily congruent via \bar{U}.