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Subdominant Eigenvalues for Stochastic Matrices with Given Column Sums

Kirkland, Steve (2009) Subdominant Eigenvalues for Stochastic Matrices with Given Column Sums. Electronic Journal of Linear Algebra, 18. pp. 784-800. ISSN 1081-3810

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For any stochastic matrix A of order n, denote its eigenvalues as λ1(A), . . . ,λn(A), ordered so that 1 = |λ1(A)| ≥ |λ2(A)| ≥ . . . ≥ |λn(A)|. Let cT be a row vector of order n whose entries are nonnegative numbers that sum to n. Define S(c), to be the set of n × n row-stochastic matrices with column sum vector cT . In this paper the quantity λ(c) = max{|λ2(A)||A ∈ S(c)} is considered. The vectors cT such that λ(c) < 1 are identified and in those cases, nontrivial upper bounds on λ(c) and weak ergodicity results for forward products are provided. The results are obtained via a mix of analytic and combinatorial techniques.

Item Type: Article
Keywords: Stochastic matrix; Subdominant eigenvalue; Bipartite graph; Hamilton Institute.
Academic Unit: Faculty of Science and Engineering > Research Institutes > Hamilton Institute
Faculty of Science and Engineering > Mathematics and Statistics
Item ID: 1896
Depositing User: Hamilton Editor
Date Deposited: 24 Mar 2010 16:20
Journal or Publication Title: Electronic Journal of Linear Algebra
Publisher: ILAS - International Linear Algebra Society
Refereed: Yes

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