  In this paper we study the capacity achieving input covariance matrices of a
single user multiantenna channel based solely on the group of symmetries of its
matrix of propagation coefficients. Our main result, which unifies and improves
the techniques used in a variety of classical capacity theorems, uses the Haar
(uniform) measure on the group of symmetries to establish the existence of a
capacity achieving input covariance matrix in a very particular subset of the
covariance matrices. This result allows us to provide simple proofs for old and
new capacity theorems. Among other results, we show that for channels with two
or more standard symmetries, the isotropic input is optimal. Overall, this
paper provides a precise explanation of why the capacity achieving input
covariance matrices of a channel depend more on the symmetries of the matrix of
propagation coefficients than any other distributional assumption.
