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Acta Mathematicae Applicatae Sinica, English Series 2011, Vol. 27 Issue (3) :503-508    DOI: 10.1007/s10255-009-9082-5
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Estimates for Eigenvalues of Stochastic Matrices
Jia-li Zhou1,2, MinWu3, Shu-you Zhang2, Guo-ping Yang4
1. Department of Mathematics, Zhejiang University of Technology, Zhejiang 310023, China;
2. State Key Lab of CAD & CG, Zhejiang University, Zhejiang 310027, China;
3. Department of Mathematics, Zhejiang University of Science & Technology, Zhejiang 310023, China;
4. College of Information Technology, Zhejiang Chinese Medical University, Zhejiang 310053, China
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Abstract It is well-known that the eigenvalues of stochastic matrices lie in the unit circle and at least one of them has the value one. Let {1, r2, · · · , rN} be the eigenvalues of stochastic matrix X of size N × N. We will present in this paper a simple necessary and sufficient condition for X such that |rj| < 1, j = 2, · · ·,N. Moreover, such condition can be very quickly examined by using some search algorithms from graph theory.  
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KeywordsDirected graph   eigenvalues   spectral radius   stochastic matrix     
Abstract: It is well-known that the eigenvalues of stochastic matrices lie in the unit circle and at least one of them has the value one. Let {1, r2, · · · , rN} be the eigenvalues of stochastic matrix X of size N × N. We will present in this paper a simple necessary and sufficient condition for X such that |rj| < 1, j = 2, · · ·,N. Moreover, such condition can be very quickly examined by using some search algorithms from graph theory.  
KeywordsDirected graph,   eigenvalues,   spectral radius,   stochastic matrix     
Received: 2009-04-29;
Fund:

Supported by grants from Science & Technology Pillar Program of Zhejiang Province (No. 2008C21084, No. 2009C31120, No. 2009C34006), Key Industrial Projects of Major Science & Technology Projects of Zhejiang Province (No. 2009C11023) and Foundation of Zhejiang Educational Committee (No. Y200804427).

Cite this article:   
.Estimates for Eigenvalues of Stochastic Matrices[J]  Acta Mathematicae Applicatae Sinica, English Serie, 2011,V27(3): 503-508
URL:  
http://www.applmath.com.cn/jweb_yysxxb_en/EN/10.1007/s10255-009-9082-5      或     http://www.applmath.com.cn/jweb_yysxxb_en/EN/Y2011/V27/I3/503
 
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