Matrix roots of imprimitive irreducible nonnegative matrices

TL;DR

This paper characterizes and classifies matrix pth-roots of imprimitive irreducible nonnegative matrices using advanced mathematical theories, providing a detailed Jordan form description and preliminary results for reducible matrices.

Contribution

It introduces a comprehensive classification of matrix pth-roots for a specific class of nonnegative matrices, extending existing theory with new characterizations.

Findings

Complete characterization of matrix pth-roots for imprimitive irreducible nonnegative matrices.

Description of roots in terms of Jordan canonical form.

Initial results on roots of reducible matrices.

Abstract

Using matrix function theory, Perron-Frobenius theory, combinatorial matrix theory, and elementary number theory, we characterize, classify, and describe in terms of the Jordan canonical form the matrix pth-roots of imprimitive irreducible nonnegative matrices. Preliminary results concerning the matrix roots of reducible matrices are provided as well.