  Asymptotic distribution for the proportional covariance model under
multivariate normal distributions is derived. To this end, the parametrization
of the common covariance matrix by its Cholesky root is adopted. The
derivations are made in three steps. First, the asymptotic distribution of the
maximum likelihood estimators of the proportionality coefficients and the
Cholesky inverse root of the common covariance matrix is derived by finding the
information matrix and its inverse. Next, the asymptotic distributions for the
case of the Cholesky root of the common covariance matrix and finally for the
case of the common covariance matrix itself are derived using the multivariate
$\delta$-method. As an application of the asymptotic distribution derived here,
a hypothesis for homogeneity of covariance matrices is considered.
