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.　©　2003　Elsevier　Inc.　All　rights　reserved.