This paper is concerned with discovering linear functional relationships among
$k$ $p$-variate populations with mean vectors $\vmu_{i}$, $i=1,\ldots ,k$ and a
common covariance matrix $\Sigma$. We consider a linear functional relationship
to be one in which each of the specified $r$ mean vectors, for example,
$\vmu_{1}, \ldots, \vmu_{r}$ are expressed as linear functions of the remainder
mean vectors $\vmu_{r+1}, \ldots, \vmu_{k}$. This definition differs from the
classical linear functional relationship, originally studied by Anderson [1],
Fujikoshi [8] and others, in that there are $r$ linear relationships among $k$
mean vectors without any specification of $k$ populations. To derive our linear
functional relationship, we first obtain a likelihood test statistic when the
covariance matrix $\Sigma$ is known. Second, the asymptotic distribution of the
test statistic is studied in a high-dimensional framework. Its accuracy is
examined by simulation.