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A generalized Burg technique has been developed recently by Burg, Luenberger, and Wegner for maximum likelihood estimation of structured covariance matrices. In this correspondence, the unique solution for the positive definite estimate over a class of nonnegative definite, symmetric matrices with known eigenvectors is presented. This solution coincides with the Karhunen-Loève expansion, and for the class of circulant matrices can be interpreted in terms of periodograms. For stationary processes and infinitely large sample size, it is shown that the sequence of optimal covariance matrices among the class of circulant matrices is asymptotically equivalent to the sequence of true covariance matrices as the observation length approaches infinity.