Left Eigenvector of a Stochastic Matrix
Sylvain Lavalle´e
DOI: 10.4236/apm.2011.14023   PDF    HTML     5,266 Downloads   12,764 Views   Citations


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 ith iprincipal minor of N=MIn , 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 .

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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.

Conflicts of Interest

The authors declare no conflicts of interest.


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