A Short Note on Kronecker Square Roots

TL;DR

This paper refines the characterization of Kronecker square roots of matrices by expressing it solely in terms of rank, simplifying previous conditions and extending applicability over fields with characteristic not equal to 2.

Contribution

It reformulates the existing matrix characterization in terms of rank alone, offering a more streamlined criterion for Kronecker square roots.

Findings

Characterization of Kronecker square roots using rank only

Applicable over fields with characteristic not equal to 2

Simplifies previous symmetry-based conditions

Abstract

The results of [I. Ojeda, Amer. Math. Monthly, 122, pp 60--64] provides a characterization of Kronecker square roots of matrices in terms of the symmetry and rank of the block vec matrix (rearrangement matrix). In this short note we reformulate the characterization in terms of rank only by considering an alternative to the block vec matrix, provided that the characteristic of the underlying field is not equal to 2.