On eigenvalue decomposition estimators of centro-symmetric covariance matrices

Abstract

This paper is focused on estimators, both batch and adaptive, of the eigenvalue decomposition (EVD) of centro-symmetric (CS) covariance matrices. These estimators make use of the property that eigenvectors and eigenvalues of such structured matrices can be estimated via two decoupled eigensystems. As a result, the number of operations is roughly halved, and moreover, the statistical properties of the estimators are improved. After deriving the asymptotic distribution of the EVD estimators, the closed-form expressions of the asymptotic bias and covariance of the EVD estimators are compared to those obtained when the CS structure is not taken into account. As a by-product, we show that the closed-form expressions of the asymptotic bias and covariance of the batch and adaptive EVD estimators are very similar provided that the number of samples is replaced by the inverse of the step size. Finally, the accuracy of our asymptotic analysis is checked by numerical simulations, and it is found that the convergence speed is also improved thanks to the use of the CS structure.