Supertropical matrix algebra III: Powers of matrices and generalized eigenspaces

TL;DR

This paper explores powers of supertropical matrices, revealing how characteristic polynomial coefficients influence matrix rank and leading to a Jordan-type decomposition and generalized eigenspaces in the supertropical setting.

Contribution

It introduces a Jordan-type decomposition and generalized eigenspaces for supertropical matrices, extending classical matrix theory to the supertropical algebra context.

Findings

Coefficients of the supertropical characteristic polynomial control matrix rank.

Established a Jordan-type decomposition for supertropical matrices.

Developed a generalized eigenspace decomposition for powers of supertropical matrices.

Abstract

We investigate powers of supertropical matrices, with special attention to the role of the coefficients of the supertropical characteristic polynomial (especially the supertropical trace) in controlling the rank of a power of a matrix. This leads to a Jordan-type decomposition of supertropical matrices, together with a generalized eigenspace decomposition of a power of an arbitrary supertropical matrix.