In many statistical inference problems involving two populations respectively with covariance matrices and its trace need to be estimated. In this paper, some existing estimators are shown to be equal besides different computational complexities after constructing some equivalent estimates based on the different characterizations of covariance matrix. Moreover, the unbiased and location-invariant estimators are constructed directly for positive integers. The asymptotical equivalence of two existing test statistics for testing the equality of two covariances is provided.
