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

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...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Authors Nishida, Yuki, Watanabe, Sennosuke, Watanabe, Yoshihide
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 04.10.2021
Subjects
Eigenvalues
Eigenvectors
Linear algebra
Mathematical analysis
Matrices (mathematics)
Matrix algebra
Multiplication
Orthogonality
Polynomials
Online AccessGet full text

Cover

More Information
Summary: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.
ISSN:2331-8422