Author(s)

Sylvain Lavalle´e

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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,N_n) , where N_i is the i–th iprincipal minor of N=M–I_n , where I_n is the identity matrix of dimension n. In the noncommutative case, this eigenvector is (P₁^-1,P_n^-1) , where P_i is the sum in Q《α_ij》 of the corresponding labels of nonempty paths starting from i and not passing through i in the complete directed graph associated to M .

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

S. Lavalle´e, "Left Eigenvector of a Stochastic Matrix," Advances in Pure Mathematics, Vol. 1 No. 4, 2011, pp. 105-117. doi: 10.4236/apm.2011.14023.