  In this paper we consider the trace regression model where $n$ entries or
linear combinations of entries of an unknown $m_1\times m_2$ matrix $A_0$
corrupted by noise are observed. We establish for the nuclear-norm penalized
estimator of $A_0$ introduced in \cite{KLT} a general sharp oracle inequality
with the spectral norm for arbitrary values of $n,m_1,m_2$ under an incoherence
condition on the sampling distribution $\Pi$ of the observed entries. Then, we
apply this method to the matrix completion problem. In this case, we prove that
it satisfies an optimal oracle inequality for the spectral norm, thus improving
upon the only existing result \cite{KLT} concerning the spectral norm, which
assumes that the sampling distribution is uniform. Note that our result is
valid, in particular, in the high-dimensional setting $m_1m_2\gg n$. Finally we
show that the obtained rate is optimal up to logarithmic factors in a minimax
sense.
