New Inclusion Regions Of Eigenvalues Different From 1 With Its Nonsingularity For A Stochastic Matrix

The location problem of eigenvalues different from 1 of stochastic matrices and nonsingular have important applications in many fields,such as computer-aided design,physics,Markov chain,etc,so the problems and its applications become one of the hot issues.There are a lot of research results about this problems,but the results are far from satisfactory about application of the increasingly high demand for accuracy.Therefore,it is significative to expiore new nonsingular conditions of stochastic matrices and more accurate positioning set of eigenvalue different from 1.At present,the main method to study the eigenvalue different from 1 of stochastic matrices is to find out a suitable vector d for obtaining the relatively precise inclusion regions of the eigenvalue different from 1 of stochastic matrices by combining the theory of modified matrices and other knowledge.This method is adopted still in this thesis.By uesing the modified matrix theory and ?-eigenvalue inclusion theorem,Brauer oval theorem,we find out a suitable vector d,furthermore,the nonsingular conditions and the inclusion regions of the eigenvalues different from 1 of stochastic matrices are obtained,and it is used to study the upper bounds of eigenvalue different from 1 modules of stochastic matrices and a new upper bound is obtained.