A Markov chain representation of the normalized Perron-Frobenius eigenvector. (English) Zbl 1373.15016

Summary: We consider the problem of finding the Perron-Frobenius eigenvector of a primitive matrix. Dividing each of the rows of the matrix by the sum of the elements in the row, the resulting new matrix is stochastic. We give a formula for the normalized Perron-Frobenius eigenvector of the original matrix, in terms of a realization of the Markov chain defined by the associated stochastic matrix. This formula is a generalization of the classical formula for the invariant probability measure of a Markov chain.


15A18 Eigenvalues, singular values, and eigenvectors
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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