# Tropical compound matrix identities

## TL;DR

This paper establishes new identities for compound matrices within extended tropical semirings, including analogues of classical matrix properties and a tropical Sylvester-Franke identity, impacting the understanding of eigenvalues.

## Contribution

It introduces novel identities for compound matrices in tropical algebra, extending classical matrix properties to the tropical setting.

## Key findings

- Identities analogous to conjugate matrix properties in tropical semirings
- Tropical Sylvester-Franke identity derived
- Implications for eigenvalues of tropical matrices

## Abstract

We prove identities on compound matrices in extended tropical semirings. Such identities include analogues to properties of conjugate matrices, powers of matrices and~$\adj(A)\det(A)^{ -1}$, all of which have implications on the eigenvalues of the corresponding matrices. A tropical Sylvester-Franke identity is provided as well.