  We completely describe the equilibrium states of a class of potentials over
the full shift which includes Falconer's singular value function for affine
iterated function systems with invertible affinities. We show that the number
of distinct ergodic equilibrium states of such a potential is bounded by a
number depending only on the dimension, answering a question of A.
K\"aenm\"aki. We prove that all such equilibrium states are fully supported and
satisfy a Gibbs inequality with respect to a suitable subadditive potential. We
apply these results to demonstrate that the affinity dimension of an iterated
function system with invertible affinities is always strictly reduced when any
one of the maps is removed, resolving a folklore open problem in the dimension
theory of self-affine fractals. We deduce a natural criterion under which the
Hausdorff dimension of the attractor has the same strict reduction property.
