随机矩阵非1特征值的新包含区域及其应用

New Inclusion Areas of Eigenvalues Different from 1 for Stochastic Matrix and Their Applications

随机矩阵在许多领域都有重要应用,而这些应用很多都与它的非1特征值有关,所以对随机矩阵的非1特征值进行定位十分有意义.应用修正矩阵理论和Gersgorin型及Brauer型矩阵特征值包含区域,获得了随机矩阵非1特征值新的Gersgorin型和Brauer型特征值包含区域及其非奇异的充分条件.数值算例说明,所得的包含区域比一些已有的包含区域更精确且能用其更好地估计随机矩阵的谱隙,从而对现有文献进行了有益补充.

It is of great significance to localize the eigenvalues different from 1 of stochastic matrix because many applications of stochastic matrix are closely related to it.By using the modified matrix theory,the Gersgorin type eigenvalues inclusion region and the Brauer type eigenvalues inclusion region,we obtained a new Gersgorin type and a new Brauer type eigenvalues inclusion region of eigenvalues different from 1 and their non-singularity sufficient conditions of stochastic matrix.The numerical examples showed that the new obtained inclusion regions were more accurate than some already existing inclusion regions and could be used to better estimate the spectral gap of the stochastic matrix.So the research provides a useful supplement to the existing literature to date.