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.
The eigenproblem of a circulant matrix in max-min algebra is investigated. Complete characterization of the eigenspace structure of a circulant matrix is given by describing all possible types of eigenvectors in detail.
A matrix A in (max,min)-algebra (fuzzy matrix) is called weakly robust if Ak⊗x is an eigenvector of A only if x is an eigenvector of A. The weak robustness of fuzzy matrices are studied and its properties are proved. A characterization of the weak robustness of fuzzy matrices is presented and an O(n2) algorithm for checking the weak robustness is described.
In max-min algebra the standard pair of operations plus and times is replaced by the pair of operations maximum and minimum, respectively. A max-min matrix A is called strongly robust if the orbit x,A⊗x,A2⊗x,… reaches the greatest eigenvector with any starting vector. We study a special type of the strong robustness called the strong \textit{\textbf{X}}-robustness, the case that a starting vector is limited by a lower bound vector and an upper bound vector. The equivalent condition for the strong \textit{\textbf{X}}-robustness is introduced and efficient algorithms for verifying the strong \textit{\textbf{X}}-robustness is described. The strong \textit{\textbf{X}}-robustness of a max-min matrix is extended to interval vectors \textit{\textbf{X}} and interval matrices \textit{\textbf{A}} using for-all-exists quantification of their interval and matrix entries. A complete characterization of AE/EA strong \textit{\textbf{X}}-robustness of interval circulant matrices is presented.
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.
A matrix A is said to have \mbox{\boldmathX}-simple image eigenspace if any eigenvector x belonging to the interval \boldmathX={x:x−−≤x≤x¯¯¯} containing a constant vector is the unique solution of the system A⊗y=x in \mbox{\boldmathX}. The main result of this paper is an extension of \mbox{\boldmathX}-simplicity to interval max-min matrix \boldmathA={A:A−−≤A≤A¯¯¯¯} distinguishing two possibilities, that at least one matrix or all matrices from a given interval have \mbox{\boldmathX}-simple image eigenspace. \mbox{\boldmathX}-simplicity of interval matrices in max-min algebra are studied and equivalent conditions for interval matrices which have \mbox{\boldmathX}-simple image eigenspace are presented. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability) that have been studied in max-min and max-plus algebras.