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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2020, Vol. 37 ›› Issue (6): 771-780.doi: 10.3969/j.issn.1005-3085.2020.06.010

• • 上一篇    

随机矩阵特征值新盖尔型包含集

朱   艳1,   周宝星2,   李耀堂2   

  1. 1- 昆明学院数学学院，昆明  650214
    2- 云南大学数学与统计学院，昆明  650091
  • 收稿日期:2018-01-04 接受日期:2020-07-28 出版日期:2020-12-15 发布日期:2021-02-15
  • 基金资助:
    国家自然科学基金 (11861077)；云南省教育厅科学研究基金 (2011Y011)；云南省科技厅面上项目 (2019FH001-078)；
    昆明学院人才引进项目 (YJL20019).

A New Ger\v{s}gorin-type Eigenvalue Localization Set for Stochastic Matrices

ZHU Yan1,   ZHOU Bao-xing2,   LI Yao-tang2   

  1. 1- School of Mathematics, Kunming University, Kunming 650214
    2- School of Mathematics and Statistics, Yunnan University, Kunming 650091
  • Received:2018-01-04 Accepted:2020-07-28 Online:2020-12-15 Published:2021-02-15
  • Supported by:
    The National Natural Science Foundation of China (11861077); the Foundation of Edcucation Commission of Yunnan Province (2011Y011); the Natural Science Foundation of Yunnan Provincial Department of Science and Technology (2019FH001-078); the Research Fund of Kunming University (YJL20019).

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摘要： 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．

关键词: 随机矩阵, 特征值, 非负矩阵, 盖尔圆盘

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

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