A canonical form for nonderogatory matrices under unitary similarity

TL;DR

This paper introduces canonical forms for nonderogatory matrices and certain matrix pairs under unitary similarity, characterized by graph structures with no cycles, advancing classification methods in matrix theory.

Contribution

It provides new canonical forms for nonderogatory matrices and matrix pairs with one having distinct eigenvalues, described via graph-based structures.

Findings

Canonical forms for nonderogatory matrices under unitary similarity.

Canonical forms for matrix pairs with one matrix having distinct eigenvalues.

Graph structures characterize the types of these canonical forms.

Abstract

A square matrix is nonderogatory if its Jordan blocks have distinct eigenvalues. We give canonical forms (i) for nonderogatory complex matrices up to unitary similarity and (ii) for pairs of complex matrices up to similarity, in which one matrix has distinct eigenvalues. The types of these canonical forms are given by undirected and, respectively, directed graphs with no undirected cycles.