On semigroups of matrices with nonnegative diagonals

TL;DR

This paper provides a concise proof of a recent theorem about the structure of irreducible matrix groups with nonnegative diagonals and explores conditions under which such semigroups are diagonally similar to nonnegative matrices.

Contribution

It offers a simplified proof of a key result and investigates the broader problem of diagonal similarity to nonnegative matrix semigroups.

Findings

Irreducible groups with nonnegative diagonals are diagonally similar to nonnegative monomial matrices.

Conditions are identified under which matrix semigroups are diagonally similar to nonnegative matrices.

The paper clarifies the structure of matrix groups and semigroups with nonnegative diagonal entries.

Abstract

We give a short proof of a recent result by Bernik, Mastnak, and Radjavi, stating that an irreducible group of complex matrices with nonnegative diagonal entries is diagonally similar to a group of nonnegative monomial matrices. We also explore the problem when an irreducible matrix semigroup in which each member is diagonally similar to a nonnegative matrix is diagonally similar to a semigroup of nonnegative matrices.