Inverse M Nature Of The Inverse Z-matrices And Related Results

Inverse M-matrices and Inverse Z-matrices are important matrix and there are many application involving inverse M-matrices and inverse Z-matrices, especially in biology, physics and mathematics. Because of valuable application of inverse M-matrices, the general properties of inverse M-matrices and inverse Z-matrices cause many interestings of studying. For M-matrices, a large number of properties and characterizations exist. However, classes of inverse M-matrices and inverse Z-matrices need to be studied furtherly. In this paper, we study a class of inverse M-matrices and inverse Z-matrices of tree structure, which have much value in application and theory and matrices associated with the original matrices, for exmple Schur complements, Perron complements, ect.In chapter two, we study a class of inverse M-matrices and inverse Z-matrices. Graph theory is used to discuss the structure and properties of inverse M-matrices and inverse Z-matrices. Sufficient and necessary conditions for a matrix to be an inverse M-matrix and inverse Z-matrix are given.In chapter three, in order to distinct N 0?matrices that defined by deferent references, we will use the following notions. A matrix is called N₀²-matrix if the matrix is nonpositive and its all-principal minors are nonpositive. We call L n=1-matrices for N₀¹-matrices. Meyer introduced the concept of the Perron complement of a nonnegative and irreducible matrix in 1989 and used it to construct an algorithm for computing the stationary distribution vector for Markov chains. We extend the Perron complements of nonnegative and irreducible matrices to the Perron complements of nonpositive and irreducible matrices. In this chapter, we show that Perron complements of N₀²-matrices are N₀²-matrices. We also demonstrate the Perron complements of inverse N₀¹-matrices are inverse N₀¹-matrices with certain restriction. In addition, we give some related inequalities about principal submatrices of inverse N₀¹-matrices.In chapter four, as is known, the Schur complements of diagonally dominant matrices are diagonally dominant and the same is true of strictly doubly diagonally...