Let ‖ A ‖ F = tr ( A ′ A ) and ‖ A ‖ 2 = λ 1 / 2 1 ( A ′ A ) denote the Frobenius norm and the spectral norm, respectively, of a matrix A . Let H$_{0}$ be the null hypothesis h$_{1}$ = ... = h$_{r}$ = 0 , and let H$_{1}$ be any of the following alternatives: H$_{1}$ : h$_{j}$ > 0 for some j = 1 , ..., r, or H$_{1}$ : Σ = σ $^{2}$I$_{p}$ , or H$_{1}$ : { Σ : ‖ Σ − σ $^{2}$I$_{p}$ ‖ F > ε$_{n,p}$ } , or H$_{1}$ : { Σ : ‖ Σ − σ $^{2}$I$_{p}$ ‖ 2 > ε$_{n,p}$ } , where ε$_{n,p}$ is a positive constant that may depend on n and p . Proposition 5 For specified σ $^{2}$, consider tests of H$_{0}$ against H$_{1}$ that are invariant with respect to the left orthogonal transformations of the data X = [ X$_{1}$, ..., X$_{n}$ ] . For any such test, there exists a test based on λ with the same power function. Similarly, for unspecified σ $^{2}$, consider tests that, in addition, are invariant with respect to multiplication of the data X by non-zero constants. For any such test, there exists a test based on µ with the same power function. Examples of the former tests include the tests of H$_{0}$ : Σ = I studied in Chen et al (2010) and Cai and Ma (2012). An example of the latter test is the test of sphericity studied in Chen et al (2010). The tests studied in Chen et al (2010) and Cai and Ma (2012) are invariant, although they are not λ - or µ -based. For r = 1 , OMH show that the asymptotic power envelopes are closely ap- proached by the asymptotic powers of the λ - and µ -based likelihood ratio tests. Our goal here is to explore the asymptotic power of those likelihood ratio tests for r > 1. Unfortunately, as r grows, it becomes increasingly difficult to compute the asymptotic critical values for the likelihood ratio tests by simulation. For ex- ample, r = 2 requires simulating a 2-dimensional Gaussian random field with the covariance function and the mean function described in Theorem 3. For r = 2 , Figure 2 shows sections of the power envelope (dotted lines) and the power of the likelihood ratio test based on λ for various fixed values of h$_{1}$$_{/}$√$_{c}$ under the alternative. Figure 3 shows the same plots for the tests based on µ. To enhance 14