This paper proposes a new test for testing the equality of two covariance matrices Σ1 and Σ2 in the high-dimensional setting and investigates its theoretical and numerical properties. The limiting null distribution of the test statistic is derived. The test is shown to enjoy certain optimality and to be especially powerful against sparse alternatives. The simulation results show that the test significantly outperforms the existing methods both in terms of size and power. Analysis of prostate cancer datasets is carried out to demonstrate the application of the testing procedures. When the null hypothesis of equal covariance matrices is rejected, it is often of significant interest to further investigate in which way they differ. Motivated by applications in genomics, we also consider two related problems, recovering the support of Σ1 − Σ2 and testing the equality of the two covariance matrices row by row. New testing procedures are introduced and their properties are studied. Applications to gene selection is also discussed
