  The design of the precoder the maximizes the mutual information in linear
vector Gaussian channels with an arbitrary input distribution is studied.
Precisely, the precoder optimal left singular vectors and singular values are
derived. The characterization of the right singular vectors is left, in
general, as an open problem whose computational complexity is then studied in
three cases: Gaussian signaling, low SNR, and high SNR. For the Gaussian
signaling case and the low SNR regime, the dependence of the mutual information
on the right singular vectors vanishes, making the optimal precoder design
problem easy to solve. In the high SNR regime, however, the dependence on the
right singular vectors cannot be avoided and we show the difficulty of
computing the optimal precoder through an NP-hardness analysis.
