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P. M. Cohn, “Skew Fields: Theory of General Di-vision Rings,” Encyclopedia of Mathematics and Its Applica-tions,” Cambridge University Press, Cambridge, 1995.

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• TITLE: Left Eigenvector of a Stochastic Matrix

  AUTHORS: Sylvain Lavalle´e

  KEYWORDS: Generic Stochastic Noncommutative Matrix, Commutative Matrix, Left Eigenvector Associated To The Eigenvalue 1, Skew Field, Automata

  JOURNAL NAME: Advances in Pure Mathematics, Vol.1 No.4, July 29, 2011

  ABSTRACT: We determine the left eigenvector of a stochastic matrix M associated to the eigenvalue 1 in the commutative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenvalue 0 is (N1,Nn) , where Ni is the i–th iprincipal minor of N=M–In , where In is the identity matrix of dimension n. In the noncommutative case, this eigenvector is (P1-1,Pn-1) , where Pi is the sum in Q《αij》 of the corresponding labels of nonempty paths starting from i and not passing through i in the complete directed graph associated to M .