Abstract: Stochastic matrix and its eigenvalue localization play key roles in many application fields such as computer aided geometric design, mathematical economics and Markov chain. Stochastic matrix eigenvalue problem contains mainly two aspects: providing a region which contains all eigenvalues different from 1 for stochastic matrices in the complex plane; estimating approximately the gap between the dominant eigenvalue 1 and the cluster of all other eigenvalues. In this paper, we localize and estimate the eigenvalues different from 1 of stochastic matrices and obtain the following results: first, we obtain a new and simple region which includes all eigenvalues of a stochastic matrix different from 1 by refining the Ger\v{s}gorin circle. Furthermore, an algorithm is proposed to estimate an upper bound for the spectral gap of the subdominant eigenvalue of a positive stochastic matrix. Numerical examples illustrate that the proposed results are effective.

Key words: stochastic matrix, eigenvalues, nonnegative matrices, Ger\v{s}gorin circle