Ergodicity Coefficients Defined by Vector Norms
SIAM Journal on Matrix Analysis and Applications
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Let S be an irreducible stochastic matrix of order n with left stationary vector $\pi^T,$ and let S(i) denote the principal submatrix of S formed by deleting the ith row and column. We prove that $\max_{1 \le i \le n}\pi_i||(I- S_{(i)})^{-1}||_{\infty} \le \min_{1 \le j \le n}||(I- S_{(j)})^{-1}||_{\infty},$ thus answering a question posed by Cho and Meyer. We provide an attainable lower bound on $\max_{1 \le i \le n}\pi_i||(I- S_{(i)})^{-1}||_{\infty},$ and discuss the case that equality holds in that bound.