On a question concerning condition numbers for Markov chains (Q2784400)
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scientific article; zbMATH DE number 1732290
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On a question concerning condition numbers for Markov chains |
scientific article; zbMATH DE number 1732290 |
Statements
23 April 2002
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stochastic matrix
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Markov chain
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stationary vector
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condition number
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On a question concerning condition numbers for Markov chains (English)
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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 NEWLINE\[NEWLINE\max_{1\leq i\leq n}\pi_i\|(I- S_{(i)})^{-1}\|_\infty\leq \min_{1\leq j\leq n}\|(I- S_{(j)})\|_\inftyNEWLINE\]NEWLINE is obtained for it. An attainable lower bound on NEWLINE\[NEWLINE\max_{1\leq i\leq n}\pi_i\|(I- S_{(i)})^{-1}\|_\inftyNEWLINE\]NEWLINE is provided, and the case of its equality is discussed.
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