Ergodicity Coefficients are Induced Matrix Seminorms

Giulia De Pasquale, Francesco Bullo, Maria Elena Valcher

Ergodicity coefficients are a useful algebraic tool in the study of the convergence properties of inhomogeneous Markov chains and averaging algorithms. Their study has a rich history, going back all the way to the original works by Markov. In this work we show how ergodicity coefficients are equal to certain induced matrix seminorms and the induced norm of optimally-deflated matrices. This equivalence clarifies their use as contraction factors for semicontractive dynamical systems. In particular, the Dobrushin ergodicity coefficient ${\tau}_1$ is shown to be equal to two different induced matrix seminorms. In the context of Markov chains, we provide an expression for the mixing time in terms of $\ell_\infty$ ergodicity coefficient. Finally, we show that, for primitive matrices, induced matrix seminorms minimize the efficiency gap in the estimation of the convergence factor of semicontractive dynamical systems, thus proving to be a more accurate tool compared to ergodicity coefficients.

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