Some Problems On Lattice Matrices

In this thesis,we describe some important properties of the lattice matrix, which include the solutions of linear system over the complete Heyting algebra, nilpotent matrix and nilpotent index over the distributive lattice and incline,and the standard eigenvectors of matrix over distributive lattice.It consists of six chapters.As the introduction,in Chapter 1,the background and history of lattice matrix are briefly addressed,and some notations,symbols and definitions are given in this chapter.In Chapter 2,the solutions of the linear system Ax=b over the complete Heyting algebra are studied.We will give some new necessary and sufficient conditions for the solvability of the system,the maximal(and the minimal) solution of the system and the unique solution when the system has unique solvabitity.In Chapter 3,the concepts of the linearly dependent and the linearly independent for n-dimension vector over the distributive lattice are introduced and some of its properties are discussed.Some new results are obtained.In Chapter 4 and in Chapter 5,the nilpotent matrix over the distributive lattice and incline are considered.By defining the new set of the submatrix,we can obtain a necessary and sufficient condition for an n×n nilpotent matrix to have the nipotent indexΥ(Υ≤n)for any given integerΥ.Another necessary and sufficient condition for an n×n nilpotent matrix to have the nilpotent indexΥ(Υ≤n)for any given integerΥis obtained by defining the associated directed graph of the matrix.These results solve the open problem proposed by Tan.In Chapter 6,the standard eigenvectors of matrix over distributive lattice are discussed.By using the knowledge of the graph theory and the associated directed graph of the matrix,we obtain the upper basic eigenvector of a given matrix.Finally we give an application of the M-matrix in differential equations.