Unitary and orthogonal equivalence of sets of matrices

TL;DR

This paper develops criteria based on trace identities to determine when two sets of matrices are equivalent under unitary, orthogonal, or complex orthogonal transformations, aiding classification in matrix analysis.

Contribution

It introduces new trace identity-based criteria for assessing simultaneous equivalence of matrix sets under various unitary and orthogonal transformations.

Findings

Provides explicit criteria for unitary equivalence of matrix sets.

Extends criteria to orthogonal and complex orthogonal equivalence.

Facilitates classification of matrices using trace identities.

Abstract

Two matrices $A$ and $B$ are called unitary (resp. orthogonal) equivalent if $AU=VB$ for two unitary (resp. orthogonal) matrices $U$ and $V$. Using trace identities, criteria are given for simultaneous unitary, orthogonal or complex orthogonal equivalence between two sets of matrices.