On the second-order statistics of the eigenvectors of sample covariance matrices

Eigenvectors of sample covariance matrices are used in a variety of statistical signal processing problems. The second-order statistics of these eigenvectors are needed to compute the variance of estimates based on these eigenvectors. Formulas for the second-order statistics of the eigenvectors have been derived in the statistical literature and are widely used. We point out a discrepancy between the statistics observed in numerical simulations and the theoretical formulas, due to the nonuniqueness of the definition of eigenvectors. We present two ways to resolve this discrepancy. The first involves modifying the theoretical formulas to match the computational results. The second involved a simple modification of the computations to make them match existing formulas.