  We analyze the global convergence of the power iterates for the computation
of a general mixed-subordinate matrix norm. We prove a new global convergence
theorem for a class of entrywise nonnegative matrices that generalizes and
improves a well-known results for mixed-subordinate $\ell^p$ matrix norms. In
particular, exploiting the Birkoff--Hopf contraction ratio of nonnegative
matrices, we obtain novel and explicit global convergence guarantees for a
range of matrix norms whose computation has been recently proven to be NP-hard
in the general case, including the case of mixed-subordinate norms induced by
the vector norms made by the sum of different $\ell^p$-norms of subsets of
entries. Finally, we use the new results combined with hypercontractive
inequalities to prove a new lower bound on the logarithmic Sobolev constant of
a Markov chain.
