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On perturbations of non-diagonalizable stochastic matrices of order 3. (English) Zbl 1430.15027

Given a stochastic \(3\times 3\) matrix which is non-diagonalizable (hence has eigenvalues \(1\) and \(\lambda\)), it is proved that it is possible to perturb this matrix to obtain a new stochastic matrix whose eigenspace corresponding to the eigenvalue \(1\) is the same and with three different eigenvalues \(1\), \(\mu\), \(\nu\) such that \(\mu\), \(\nu\) are arbitrarily close to \(\lambda\).

MSC:

15B51 Stochastic matrices
15A18 Eigenvalues, singular values, and eigenvectors
47A55 Perturbation theory of linear operators
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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