The Estimation Of Maximum Eigenvalue Of Nonnegative Irreducible Matrix

Nonnegative matrix theory has been one of the most active research areas in linear algebra.It was widely used in the areas of numerical Analysis,computer science,dynamic programming and population statistics.This article based on Perron-Frobenius theorem of nonnegative irreducible matrix expansionly study its maximum characteristic value.The main content of this paper is as follows:In chapter 1,we introduce the developments and background of the spectral radius of nonnegative irreducible matrix,as well as present situations of the research at home and abroad.In chapter 2,we give some basic definitions,basic theorems and some useful algorithm of nonnegative irreducible matrices.In chapter 3,we introduces a new kind of estimate the maximum eigenvalue of nonnegative irreducible matrix methods,we obtained a new method which estimate the maximum eigenvalue of nonnegative irreducible matrix by constructing two matrices,B-?A² + ?₁I+ A?^n-1 and C=?A²-?₂I+ A?^n-1,then taking advantage of P-F theroy,sum of rows and columns of constructed matrices,also introduced a useful limit estimation formula in theroy.And numerical example is given to show that the results obtained by the new method are more accurate than those obtained by literatures[16-19].In chapter 4,on the basis of the new Nonnegative Irreducible matrix,the bounds of the upper and lower bounds for the maximum eigenvalue of Nonnegative Irreducible matrices are derived:It is proved theoretically the limit estimate formula.