Further comments on "An alternate derivation of the maximum likelihood estimate of a covariance matrix"

The alternate derivation of the maximum likelihood estimate (MLE) of a covariance matrix, as offered by Nitzberg still contains certain problems, albeit his recent correspondence has assisted considerably in understanding the approach. A major point of difference centers around the last few steps of the alternate proof where a transition is attempted from the MLE of certain eigenvalues to the associated matrix, and thence to the MLE of the covariance matrix. The transition is effected by first "recognizing" that (for Normal statistics) the MLE of the covariance matrix is unique, and then using properties dependent on the uniqueness to derive the MLE. Rigorous development requires that the uniqueness, or at least any necessary structural characterizations, be first ascertained. A rough proof in this regard is presented here. An earlier reference containing the alternate derivation presented by Nitzberg is also provided.