By max-plus algebra we mean the set of reals R equipped with the operations a⊕b=max{a,b} and a⊗b=a+b for a,b∈R. A vector x is said to be a generalized eigenvector of max-plus matrices A,B∈R(m,n) if A⊗x=λ⊗B⊗x for some λ∈R. The investigation of properties of generalized eigenvectors is important for the applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. In this paper the properties of matrices and vectors with inexact (interval) entries are studied and complete solutions of the controllable, the tolerable and the strong generalized eigenproblem in max-plus algebra are presented. As a consequence of the obtained results, efficient algorithms for checking equivalent conditions are introduced.
Fuzzy algebra is a special type of algebraic structure in which classical addition and multiplication are replaced by maximum and minimum (denoted ⊕ and ⊗ , respectively). The eigenproblem is the search for a vector x (an eigenvector) and a constant λ (an eigenvalue) such that A⊗x=λ⊗x , where A is a given matrix. This paper investigates a generalization of the eigenproblem in fuzzy algebra. We solve the equation A⊗x=λ⊗B⊗x with given matrices A,B and unknown constant λ and vector x . Generalized eigenvectors have interesting and useful properties in the various computational tasks with inexact (interval) matrix and vector inputs. This paper studies the properties of generalized interval eigenvectors of interval matrices. Three types of generalized interval eigenvectors: strongly tolerable generalized eigenvectors, tolerable generalized eigenvectors and weakly tolerable generalized eigenvectors are proposed and polynomial procedures for testing the obtained equivalent conditions are presented.