  The classic likelihood ratio test for testing the equality of two covariance
matrices breakdowns due to the singularity of the sample covariance matrices
when the data dimension $p$ is larger than the sample size $n$. In this paper,
we present a conceptually simple method using random projection to project the
data onto the one-dimensional random subspace so that the conventional methods
can be applied. Both one-sample and two-sample tests for high-dimensional
covariance matrices are studied. Asymptotic results are established and
numerical results are given to compare our method with state-of-the-art methods
in the literature.
