Independence and orthogonality of algebraic eigenvectors over the max-plus algebra

TL;DR

This paper explores the properties of algebraic eigenvectors in max-plus algebra, demonstrating their independence and orthogonality under certain conditions, thus extending classical linear algebra concepts to this algebraic setting.

Contribution

It establishes the independence and orthogonality properties of algebraic eigenvectors in max-plus algebra, paralleling classical linear algebra results.

Findings

Algebraic eigenvectors with distinct eigenvalues are linearly independent for generic matrices.

Algebraic eigenvectors are orthogonal for symmetric matrices with distinct eigenvalues.

Properties mirror classical eigenvector behaviors in a max-plus algebra context.

Abstract

The max-plus algebra $\mathbb{R}\cup \{-\infty \}$ is a semiring with the two operations: addition $a \oplus b := \max(a,b)$ and multiplication $a \otimes b := a + b$. Roots of the characteristic polynomial of a max-plus matrix are called algebraic eigenvalues. Recently, algebraic eigenvectors with respect to algebraic eigenvalues were introduced as a generalized concept of eigenvectors. In this paper, we present properties of algebraic eigenvectors analogous to those of eigenvectors in the conventional linear algebra. First, we prove that for generic matrices algebraic eigenvectors with respect to distinct algebraic eigenvalues are linearly independent. We further prove that for symmetric matrices algebraic eigenvectors with respect to distinct algebraic eigenvalues are orthogonal to each other.