  This paper considers testing a covariance matrix $\Sigma$ in the high
dimensional setting where the dimension $p$ can be comparable or much larger
than the sample size $n$. The problem of testing the hypothesis
$H_0:\Sigma=\Sigma_0$ for a given covariance matrix $\Sigma_0$ is studied from
a minimax point of view. We first characterize the boundary that separates the
testable region from the non-testable region by the Frobenius norm when the
ratio between the dimension $p$ over the sample size $n$ is bounded. A test
based on a $U$-statistic is introduced and is shown to be rate optimal over
this asymptotic regime. Furthermore, it is shown that the power of this test
uniformly dominates that of the corrected likelihood ratio test (CLRT) over the
entire asymptotic regime under which the CLRT is applicable. The power of the
$U$-statistic based test is also analyzed when $p/n$ is unbounded.
