Independence and orthogonality of algebraic eigenvectors over the max-plus algebra
Yuki Nishida, Sennosuke Watanabe, Yoshihide Watanabe
TL;DR
This work addresses eigenstructure in the max-plus (tropical) setting, focusing on algebraic eigenvectors defined via the roots of the max-plus characteristic polynomial $\chi_A(t)$ and the associated algebraic eigenspaces $W(A,\lambda)$. It introduces a perturbation-based extension via $A(\bm{\zeta};\delta)$ to apply the theory to all $n\times n$ matrices, and connects $W(A,\lambda)$ to the adjugate $\Gamma(A,\lambda) = \mathrm{adj}(A \oplus \lambda \otimes E_n)$. The main results show that for distinct algebraic eigenvalues, the spaces $W(A,\lambda)$ intersect trivially, and under a mild disjoint-circuit condition the union of basis vectors across eigenvalues spans the sum of all algebraic eigenspaces; for max-plus symmetric matrices, these eigenvectors are tropical-orthogonal. Collectively, these findings advance the max-plus spectral theory by clarifying independence and orthogonality of algebraic eigenvectors and by providing constructive bases for algebraic eigenspaces through graph-analytic and adjugate-based constructs.
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.