Derive the constrained mles for the microk and sigma show

Question

Consider the multivariate Gaussian model X|G = k ∼ N(µk, Σ), with the additional restriction that rank

Derive the constrained MLEs for the µk and Σ. Show that the Bayes classification rule is equivalent to classifying in the reduced subspace computed by LDA (Hastie and Tibshirani 1996).