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On a question concerning condition numbers for Markov chains. (English) Zbl 1013.15005
An irreducible stochastic matrix $S$ of order $n$ with stationary vector $\pi^T$, and the principal submatrix $S_{(i)}$ formed by deleting the $i$th row and column of $S$ is considered. The relation $$\max_{1\le i\le n}\pi_i\|(I- S_{(i)})^{-1}\|_\infty\le \min_{1\le j\le n}\|(I- S_{(j)})\|_\infty$$ is obtained for it. An attainable lower bound on $$\max_{1\le i\le n}\pi_i\|(I- S_{(i)})^{-1}\|_\infty$$ is provided, and the case of its equality is discussed.

15B51Stochastic matrices
15A45Miscellaneous inequalities involving matrices
65F35Matrix norms, conditioning, scaling (numerical linear algebra)
60J10Markov chains (discrete-time Markov processes on discrete state spaces)
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