  In traditional Ergodic Optimization, one seeks to maximize Birkhoff averages.
The most useful tool in this area is the celebrated Ma\~n\'e Lemma, in its
various forms. In this paper, we prove a non-commutative Ma\~n\'e Lemma, suited
to the problem of maximization of Lyapunov exponents of linear cocycles or,
more generally, vector bundle automorphisms. More precisely, we provide
conditions that ensure the existence of an extremal norm, that is, a Finsler
norm with respect to which no vector can be expanded in a single iterate by a
factor bigger than the maximal asymptotic expansion rate. These conditions are
essentially irreducibility and sufficiently strong fiber bunching. Therefore we
extend the classic concept of Barabanov norm, which is used in the study of the
joint spectral radius. We obtain several consequences, including sufficient
conditions for the existence of Lyapunov maximizing sets.
