Asymptotic Nearness of Stochastic and Doubly-Stochastic Matrices
نویسنده
چکیده
We prove that the set of n×n positive (row-)stochastic matrices and the corresponding set of doubly-stochastic matrices are asymptotically close. More specifically, random matrices within each of these classes are arbitrarily close in sufficiently high dimensions. AMS 2000 subject classifications:Primary 15A51,15A52,15A60, Stochastic Matrices. Let Sn denote the set of n × n stochastic matrices with positive entries, and define 1 as the n× 1 vector of ones. By definition, any A ∈ Sn satisfies A1 = 1. (1) The Perron-Frobenius theory of positive matrices Seneta (1981); Horn and Johnson (1991) provides a comprehensive characterization of the spectrum of such matrices. Namely, denoting {λi}i=1 as the eigenvalues, we know that 1. λ1 = 1 is the unique eigenvalue of A with maximum modulus; 2. λ1 corresponds to positive right and left eigenvectors v1 and u1 where
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