UMP invariant tests for a generalized linear model

For a generalized normal linear model in which the covariance matrix [Sigma] is positive definite symmetric, UMP invariant test procedures for some kinds of linear hypotheses are derived by transforming the model by an orthogonal matrix L, consisting of orthonormal eigenvectors of [Sigma] as the columns vectors. Here it is assumed that [Sigma] contains unknown elements but has a certain structure making all the elements of L known. A sufficient condition for this assumption is also obtained to examine whether the covariance matrix [Sigma] has such a form.