  Motivated by problems arising in random sampling of trigonometric
polynomials, we derive exponential inequalities for the operator norm of the
difference between the sample second moment matrix $n^{-1}U^*U$ and its
expectation where $U$ is a complex random $n\times D$ matrix with independent
rows. These results immediately imply deviation inequalities for the largest
(smallest) eigenvalues of the sample second moment matrix, which in turn lead
to results on the condition number of the sample second moment matrix. We also
show that trigonometric polynomials in several variables can be learned from
$const \cdot D \ln D$ random samples.
