| From the view of the system structure of mathematical point, object of mathematical research into three basic structure: orderly, algebra, topology, lattice is an important structure of order and algebra, it has very close links with mathematics, topology and other modern mathematics; From the concept of lattices appear in various areas of mathematics, lattice is very important for mathematics and other branches contact and applied. In computer science, the field of cryptology, graph theory, functional analysis and switching theory have direct application of the lattice. Distributive lattice is a very important grid of lattice theory, and matrix is an important tool for mathematical research and mathematical applications. So we study on the matrix of the distributive lattice is more popular topics of the lattice theory study. In computer theory, category theory, algebraic topology and fuzzy mathematics has a wide range of applications. This paper studies eigenvector of matrix on distributive lattices, and gives a little of my views and conclusions from the algebraic structure, properties and calculation methods.The article is divided into three parts:The first chapter introduces research background, history, research status and innovation. It gives the required knowledge of definitions, characters and lemma. The part of the lattices gives definition of lattices, metrical, computing and the nature of the matrix on the lattices. The part of the graph theory introduces the associated graph, path, cycle and other related knowledge. The part of the power series includes power series, the definition of join-irreducible elements and decomposition theorem of join-irreducible elements. The part of the complete and completely distributive lattices includes the definition of complete and completely distributive lattices and lemma.The second part, first, describes two methods to solve standard eigenvectors of the matrix on the distributive lattices. The first method is using graph theory knowledge to solve the standard eigenvectors *x of matrix on the distributive lattices. The second method is using generalized sequence to demand standard eigenvectors of a matrix on the distributive lattices, and illustrated. Then we also discussed the solution of eigenvectors of symmetric matrix on the distributive lattices.Secondly, we discuss the algebraic structure and calculation method.of all standard eigenvectors of matrix on distributive lattices. In order to study the need of eigenvector, first we introduce decomposition theorem of matrix on the distributive lattices, point that the matrix of distributive lattices can have the similar of join-irreducible decomposition of elements of lattices. Then we use the power sequence of matrix on the distributive lattices to get formulas of all standard eigenvector. Further we obtain the solving method of the general eigenvector of matrix. For getting standard eigenvector, we gives a general calculation method of(n)A.The third part introduces the solution and property of eigenvector of matrix on the complete and completely lattice. We use the pseudocomplement to get formulas of the largest eigenvector of matrix on the complete and completely lattice, and illustrate how to use the formula for solving the largest eigenvectors. Proved that the eigenvector corresponding to the eigenvalues constitute an interval, and gives the expression of the endpoint of this interval, and when there is a unique eigenvalues of the eigenvector, we give the range of the eigenvalues satisfy. |
