Subdominant eigenvalues for stochastic matrices with given column sums

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Stephen J. Kirkland


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  = max{|λ2(A)||A ∈ S(c)} is considered. The vectors cT such that  < 1 are identified and in those cases, nontrivial upper bounds on  and weak ergodicity results for forward products are provided. The results are obtained via a mix of analytic and combinatorial techniques.

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