• Corpus ID: 238253264

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

@inproceedings{Nishida2021IndependenceAO,
  title={Independence and orthogonality of algebraic eigenvectors over the max-plus algebra},
  author={Yuki Nishida and Sennosuke Watanabe and Yoshihide Watanabe},
  year={2021}
}

• Yuki Nishida, Sennosuke Watanabe, Yoshihide Watanabe
• Published 1 October 2021
• Mathematics, Computer Science

The max-plus algebra R ∪ {−∞} is a semiring with the two operations: addition a ⊕ b := max(a, b) and multiplication a ⊗ 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.