On the characteristic polynomial of a supertropical adjoint matrix

TL;DR

This paper proves a tropical analogue of a classical linear algebra result, showing a reciprocal relationship between characteristic polynomials of matrices and their inverses in the supertropical setting.

Contribution

It confirms a recent conjecture by Niv by establishing the characteristic polynomial relation for supertropical matrices, extending classical results to tropical algebra.

Findings

Proves the tropical analogue of the reciprocal characteristic polynomial relation.

Confirms Niv's conjecture on supertropical matrices.

Extends classical linear algebra results to tropical algebra.

Abstract

Let $\chi(A)$ denote the characteristic polynomial of a matrix $A$ over a field; a standard result of linear algebra states that $\chi(A^{-1})$ is the reciprocal polynomial of $\chi(A)$. More formally, the condition $\chi^n(X) \chi^k(X^{-1})=\chi^{n-k}(X)$ holds for any invertible $n\times n$ matrix $X$ over a field, where $\chi^i(X)$ denotes the coefficient of $\lambda^{n-i}$ in the characteristic polynomial $\det(\lambda I-X)$. We confirm a recent conjecture of Niv by proving the tropical analogue of this result.