This correspondence investigates the gain in statistical performance/complexity of the adaptive estimation of the eigenvalue decomposition (EVD) of covariance matrices when the centrosymmetric (CS) structure of such matrices is utilized. After deriving the asymptotic distribution of the EVD estimators, it is shown, in particular, that the closed-form expressions for the asymptotic covariance of batch and adaptive EVD estimators are very similar, provided that the number of samples is replaced by the inverse of the step size.