On pseudo-invereses of matrices and their characteristic polynomials in supertropical algebra

TL;DR

This paper explores the properties of pseudo-inverses in supertropical algebra, focusing on their characteristic polynomials, similarity relations, and connections to matrix stabilization, revealing both classical and novel algebraic behaviors.

Contribution

It introduces the concept of pseudo-inverses in supertropical algebra, analyzes their characteristic polynomials, and establishes new properties and proofs related to matrix similarity and stabilization.

Findings

Pseudo-inverses inherit classical properties and exhibit new behaviors.

Characteristic polynomials of pseudo-inverses and similar matrices are studied.

A new proof for the identity of det(AB) and links to matrix power stabilization are provided.

Abstract

The only invertible matrices in tropical algebra are diagonal matrices, permutation matrices and their products. However, the pseudo-inverse $A^\nabla$, defined as $\frac{adj(A)}{det(A)}$, with $det(A)$ being the tropical permanent (also called the tropical determinant) of a matrix $A$, inherits some classical algebraic properties and has some surprising new ones. Defining $B$ and $B'$ to be tropically similar if $B' =A^\nabla BA$, we examine the characteristic (max-)polynomials of tropically similar matrices as well as those of pseudo-inverses. Other miscellaneous results include a new proof of the identity for $det(AB)$ and a connection to stabilization of the powers of definite matrices.