Yet another entropy power inequality with an application

In this paper we derive a generalization of the vector entropy power inequality (EPI) recently put forth in [1], which was valid only for diagonal matrices, to the full matrix case. Next, we study the problem of computing the linear precoder that maximizes the mutual information in linear vector Gaussian channels with arbitrary inputs. In particular, we transform the precoder optimization problem into a new form and, capitalizing on the newly unveiled matrix EPI, we show that some particular instances of the optimization problem can be cast in convex form, i.e., we can have an optimality certificate, which, to the best of our knowledge, had never been obtained previously.