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Zbl 1013.15005
Kirkland, S.
On a question concerning condition numbers for Markov chains. (English)
[J] SIAM J. Matrix Anal. Appl. 23, No.4, 1109-1119 (2002). ISSN 0895-4798; ISSN 1095-7162/e

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.
[Václav Burjan (Praha)]

MSC 2000:: *15A51 Stochastic matrices
15A45 Miscellaneous inequalities involving matrices
65F35 Matrix norms, etc. (numerical linear algebra)
60J10 Markov chains with discrete parameter

Keywords: stochastic matrix; Markov chain; stationary vector; condition number

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