On real algebras generated by positive and nonnegative matrices

TL;DR

This paper characterizes algebras generated by positive and nonnegative matrices, providing conditions for their structure, and solves a problem regarding dimensions of algebras generated by semi-commuting nonnegative matrices.

Contribution

It offers new descriptions of algebras generated by positive matrices and determines possible dimensions for algebras generated by two semi-commuting nonnegative matrices.

Findings

Algebras generated by strictly positive matrices are classified up to similarity.

Sufficient conditions are given for block diagonal matrix algebras to be generated by nonnegative matrices.

All realizable dimensions of algebras generated by two semi-commuting nonnegative matrices are identified.

Abstract

Algebras generated by strictly positive matrices are described up to similarity, including the commutative, simple, and semisimple cases. We provide sufficient conditions for some block diagonal matrix algebras to be generated by a set of nonnegative matrices up to similarity. Also we find all realizable dimensions of algebras generated by two nonnegative semi-commuting matrices. The last result provides the solution to the problem posed by M. Kandi\'{c}, K. \v{S}ivic (2017).