On the eigenproblem of matrices over distributive lattices

Let (L, , , ) be a complete and completely distributive lattice. A vector ξ is said to be an eigenvector of a square matrix A over the lattice L if Aξ=λξ for some λ in L. The elements λ are called the associated eigenvalues. In this paper, we obtain the maximum eigenvector of A for a given eigenvalue λ, and give some properties of the maximum matrix M(λ,ξ) in T(λ,ξ), the set of matrices with a given eigenvector ξ and eigenvalue λ. We also consider the structure of matrices which possess a given primitive eigenvector ξ and show in particular that, for any given λ in L, there is a matrix, namely M(λ,ξ), having ξ as a maximal primitive eigenvector associated with the eigenvalue λ.