Left Eigenvector of a Stochastic Matrix

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DOI: 10.4236/apm.2011.14023   PDF   HTML     4,471 Downloads   11,423 Views   Citations

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 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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