On the eigenproblem of matrices over distributive lattices

Let (L, less than or equal to, boolean OR, boolean AND) be a complete and completely distributive lattice. A vector xi is said to be an eigenvector of a square matrix A over the lattice L if Axi = lambdaxi for some lambda in L. The elements lambda are called the associated eigenvalues. In this paper, we obtain the maximum eigenvector of A for a given eigenvalue lambda, and give some properties of the maximum matrix M(lambda, xi) in T(lambda, xi), the set of matrices with a given eigenvector xi and eigenvalue lambda. We also consider the structure of matrices which possess a given primitive eigenvector xi and show in particular that, for any given lambda in L, there is a matrix, namely M(lambda,xi) having xi as a maximal primitive eigenvector associated with the eigenvalue lambda. (C) 2003 Elsevier Inc. All rights reserved.