We investigate two ergodicity coefficients ϕ∥∥​ and τn−1​,
originally introduced to bound the subdominant eigenvalues of nonnegative
matrices.
The former has been generalized to complex matrices in recent years and
several properties for such generalized version have been shown so far.
We provide a further result concerning the limit of its powers. Then we
propose a generalization of the second coefficient τn−1​ and we show
that, under mild conditions, it can be used to recast the eigenvector problem
Ax=x as a particular M-matrix linear system, whose coefficient matrix can be
defined in terms of the entries of A. Such property turns out to generalize
the two known equivalent formulations of the Pagerank centrality of a graph