Ergodicity Coefficients Defined by Vector Norms

Ergodicity coefficients for stochastic matrices determine inclusion regions for subdominant eigenvalues; estimate the sensitivity of the stationary distribution to changes in the matrix; and bound the convergence rate of methods for computing the stationary distribution. We survey results for ergodicity coefficients that are defined by $p$-norms, for stochastic matrices as well as for general real or complex matrices. We express ergodicity coefficients in the one-, two-, and infinity-norms as norms of projected matrices, and we bound coefficients in any $p$-norm by norms of deflated matrices. We show that two-norm ergodicity coefficients of a matrix $A$ are closely related to the singular values of $A$. In particular, the singular values determine the extreme values of the coefficients. We show that ergodicity coefficients can determine inclusion regions for subdominant eigenvalues of complex matrices, and that the tightness of these regions depends on the departure of the matrix from normality. In the special case of normal matrices, two-norm ergodicity coefficients turn out to be Lehmann bounds.