Properties Of Nonnegative Irreducible Matrices And Estimation Of Their Spectral Radius

Matrix is an important branch in modern mathematics and it has an advantage of being concise in dealing with complex problems, so it become an important tool in dealing with mathematical and engineering problems. Nonnegative matrix, a main cateory in matrix theory, has a vast applications in many fields like automatic control theory, operations research, linear programming theory, computational mathematics, graph theory, etc. And it has a very realistic significance to nonnegative matrices’ research. One of the most important problems is the boundary value estimation of spectral radius for nonnegative matrix.This paper mainly studies the properties of nonnegative matrix and nonnegative irreducible matrix, gets some conclusions about bounds for the spectral radius of nonnegative matrix, and make an new estimation of the spectral radius for nonnegative irreducible matrix by using these results.This paper could be divided into four chapters altogether. The main content of every chapter are as follows:In the first chapter, some symbols, basic definitions and theories which need to be used, are introduced in this paper. The nonnegative matrix, the nonnegative irreducible matrix, displacement matrix, spectral radius, the largest eigenvector concepts and some theorems related are included in this chapter.The second chapter, some equivalent conditions of nonnegative matrices irreducibility are proved by reconstructing new matrix and disproving with some already known theories. Also, the properties of structure and elements for nonnegative irreducible matrix are concluded and discussed.The third chapter, some theorems of positive matrix and nonnegative matrix bounds for the range of spectral radius are proved by using the properties of nonnegative matrices. In order to get the size of boundary value range,some analysis and comparison are carried out. Also, the results are tested and verified through specific numerical example.The fourth chapter, some range of boundary value of spectral radius for nonnegative irreducible matrix are given out by utilizing the properties of nonnegative irreducible matrix and Collate-Wielandt function. The new method for calculating the spectral radius of nonnegative irreducible matrix is obtained by using the maximum thought. It is proved to be feasible.