On the vectors associated with the roots of max-plus characteristic polynomials

Abstract

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz (2020) APPLICATIONS OF MATHEMATICSNo. 6, 785-805

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

“…In the recent paper [23], the authors have introduced algebraic eigenvectors with respect to algebraic eigenvalues under the following assumption for a matrix A:…”

“…Recently, the authors have clarified the role of the roots of the characteristic polynomial χ A (t) of a max-plus matrix A [23]. Each coefficient of the characteristic polynomial of A ∈ R n×n max corresponds to the weight of a multi-circuit in the associated graph G(A), where a multi-circuit is the union of disjoint elementary circuits in the graph.…”

“…A multi-circuit is called λ-maximal if the corresponding term attains the maximum of χ A (λ). The paper [23] was written under the assumption that for each λ ∈ R max the lengths of any two distinct λ-maximal multi-circuits are different. For an algebraic eigenvalue λ and a λ-maximal multi-circuit C, a nontrivial vector x satisfying…”

“…is called an algebraic eigenvector of A with respect to λ, where matrices A C , A \C , E C , and E \C and defined in Section 2.4. We have proved in [23] that algebraic eigenvectors possess many good properties so that we regard them as the extension of eigenvectors: (i) all eigenvectors are also algebraic eigenvectors (ii) the set of all algebraic eigenvectors with respect to an algebraic eigenvalue is a max-plus subspace, called algebraic eigenspace, and (iii) the dimension of the algebraic eigenspace does not exceed the multiplicity of the algebraic eigenvalue as a root of the characteristic polynomial.…”

“…In Section 4, we prove that algebraic eigenvectors of a max-plus symmetric matrix with respect to distinct algebraic eigenvalues are also orthogonal to each other. A max-plus symmetric matrix usually has two distinct λ-maximal multi-circuits with the same length, not satisfying the assumption in [23] noted above. This is because each multi-circuit is accompanied by the reversed one.…”