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Generalized Eigenvectors and Rayleigh bounds for tropical algebraic eigenvalues

Dariush Kiani, Hanieh Tavakolipour

Abstract

In this paper, we review the eigenpair problem in the context of tropical algebra. An important fact that has been largely overlooked in spectral theory of tropical algebra is that the tropical algebraic eigenvalues, which are obtained from the characteristic polynomial, may not correspond to any tropical eigenvector satisfying the standard eigenvalue-eigenvector equation. To resolve this, we use the tropical numerical range and define a generalized tropical eigenvalue-eigenvector relation. We define any non-zero vector satisfying this equation as a generalized tropical eigenvector. We show that a generalized tropical eigenvector always exists for any given tropical algebraic eigenvalue. We propose a computationally inexpensive method for the construction of these vectors. Additionally, we prove an upper bound for the algebraic eigenvalues of a tropical matrix, using the tropical Rayleigh quotients.

Generalized Eigenvectors and Rayleigh bounds for tropical algebraic eigenvalues

Abstract

In this paper, we review the eigenpair problem in the context of tropical algebra. An important fact that has been largely overlooked in spectral theory of tropical algebra is that the tropical algebraic eigenvalues, which are obtained from the characteristic polynomial, may not correspond to any tropical eigenvector satisfying the standard eigenvalue-eigenvector equation. To resolve this, we use the tropical numerical range and define a generalized tropical eigenvalue-eigenvector relation. We define any non-zero vector satisfying this equation as a generalized tropical eigenvector. We show that a generalized tropical eigenvector always exists for any given tropical algebraic eigenvalue. We propose a computationally inexpensive method for the construction of these vectors. Additionally, we prove an upper bound for the algebraic eigenvalues of a tropical matrix, using the tropical Rayleigh quotients.
Paper Structure (10 sections, 9 theorems, 41 equations, 1 figure)

This paper contains 10 sections, 9 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. motivation
  3. Generalized tropical eigenvector
  4. preliminaries
  5. Tropical Algebra, polynomials, and roots
  6. Tropical Matrix Operations and Characteristic Polynomial
  7. Tropical numerical range
  8. Generalized tropical eigenvector
  9. Computation of generalized tropical eigenvectors
  10. Tropical Rayleigh quotient upper bound for $\lambda_k$

Key Result

Lemma 3.2

Let $A \in \mathbb{R}_{\max}^{n \times n}$. Then the set of values $\{x^T \otimes A \otimes x : x \in \mathbb{R}_{\max}^n, \Vert x \Vert = 0\}$ is equal to $F_{\max}(A)$.

Figures (1)

  • Figure 1: Explicit formula for the generalized tropical eigenvector corresponding to the eigenvalue $\lambda$.

Theorems & Definitions (22)

  • Definition 2.1: Tropical Roots
  • Definition 2.2: Tropical Algebraic Eigenvalue
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1: Tropical numerical range
  • Lemma 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 4.1
  • proof
  • ...and 12 more