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 .