Classification of squared normal operators on unitary and Euclidean spaces

TL;DR

This paper establishes canonical forms for complex and real matrices whose squares are normal, under unitary and orthogonal similarity transformations, respectively, advancing the classification of such matrices in linear algebra.

Contribution

It introduces new canonical forms for matrices with normal squares, extending the understanding of their structure under similarity transformations.

Findings

Canonical form for complex matrices with normal squares.

Canonical form for real matrices with normal squares.

Enhanced classification of matrices based on their squared normality.

Abstract

We give a canonical form for a complex matrix, whose square is normal, under transformations of unitary similarity as well as a canonical form for a real matrix, whose square is normal, under transformations of orthogonal similarity.