Typically, optimal properties for principal components concern the simultaneous minimization of eigenvalues of certain covariance matrices which measure the goodness of an approximation. Many population criteria like total variance and generalized variance, which are increasing functions of the eigenvalues, are then minimized by the best approximator.
In other situations, the criterion may not be a monotone function of the eigenvalues. In Theorem 1, we derive a general optimal class based on the non-negative definite ordering of covariance matrices.
AMS (MOS) subject classification
Principal components statistical approximations
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