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Matrix roots and embedding conditions for three-state discrete-time Markov chains with complex eigenvalues. (English) Zbl 1504.60115

Let \(P\) be the transition matrix for a finite state discrete-time Markov chain. Then \(P\) is also called a stochastic matrix; \(P\) has nonnegative entries and all row sums equal to \(1\). \(P\) is said to have a stochastic root \(A\) if \(A\) is a stochastic matrix and there exists an integer \(m \geq 2\) such that \(P = A^m\).
Necessary and sufficient conditions for stochastic roots have been established for Markov chains with 2 states, and with 3 states if all eigenvalues are real. In this work, the latter case is settled by considering the case when two of the eigenvalues are complex conjugates (recall that the largest eigenvalue is always \(1\)).

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
15A18 Eigenvalues, singular values, and eigenvectors
15B51 Stochastic matrices
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References:

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