In certain multivariate problems involving several populations, the covariance structure of the populations is such that all covariance matrices can be diagonalized simultaneously by a fixed orthogonal transformation. In the transformed problem one has a number of independent univariate populations. Consequently certain hypotheses in the original problem become equivalent to simultaneous hypotheses on these univariate populations in the transformed model. Using this approach we propose a test procedure for testing the hypothesis of equality of covariance matrices against a certain alternative under the intraclass correlation model. The relative advantages of our procedure over that of Srivastava's procedure [6] are also discussed. Finally we indicate how the problem of testing for the equality of covariance matrices under a more general set up can be reduced to a univariate problem.