 Solution

Transform each observation to $Y_j=A^T{X}_j$. If $A^T$ has full row rank $m$, the transformed sample is multivariate normal with mean $A^T\mu$ and positive-definite covariance matrix $A^T\Sigma A$. Its sample mean is $A^T\bar{X}$ and its unbiased sample covariance matrix is $A^{T}SA$. The Hotelling test of linear hypotheses therefore usesHere $n>m$ is required. Reject for a large scaled statistic using the relevant upper F-distribution quantile. If the rows of $A^T$ are dependent, choose a basis for their row space, use that basis as the contrast matrix, and replace $m$ by its rank $r$. The null hypothesis is unchanged, but redundant contrasts must not be treated as independent dimensions. Rank zero gives no restrictions and hence no nontrivial test.