New Sets To Localize All Eigenvalues Different From 1 For Stochastic Matrices

As a special kind of nonnegative matrices, stochastic matrices have extensive application background. Many problems in mathematical economics, operations research and Markov chains and so on all have a close connection with stochastic matrices. Since the localization of all eigenvalues different from 1 is contacted close-ly with nonsingular criteria for real matrices with same nonzero row sums, both of them play important roles in the stochastic matrices theory, which have become one of the hot issues in recent years. In this thesis, we continue to study these problems, on basis of the modified matrices theory and the nonsingularity of S-SDD matrices, some new nonsingular criterias of real matrices with same nonzero row sums are given, and some new inclusion sets of all eigenvalues different from 1 for stochastic matrices are obtained. Numerical examples illustrate that the proposed results are more accurate than the results of Cvetkovic et al. [Electronic Transactions on Nu-merical Analysis,18(2004)73-80], Shen et al. [Linear Algebra and its Applications, 447(2014)74-87] and Li et al. [Linear and Multilinear Algebra,11(2015)2159-2170].