随机矩阵特征值新盖尔型包含集

A New Gersgorin-type Eigenvalue Localization Set for Stochastic Matrices

随机矩阵及其特征值问题具有广泛的应用背景,计算机辅助几何设计、数理经济学和马尔科夫链等领域都与其有着密切的联系.对随机矩阵特征值问题的研究主要集中在两个方面:在复平面上给出包含随机矩阵所有非1特征值的区域;给出随机矩阵特征值1和非1特征值之间距离的近似值估计.本文对这两方面问题进行了研究,获得了如下结果:通过选择新的参数,获得随机矩阵非1特征值新盖尔型包含区域,改进了近期一些相关成果.并由此得到估计正随机矩阵特征值1与非1特征值距离的新上界算法.最后,数值例子表明算法的优越性.

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 sgorin 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.