Eigenvector properties of Toeplitz matrices and their application to spectral analysis of time series

Abstract

This paper presents a proof for an important eigenvector property of Toeplitz matrices. The property, a variant of Caratheodory's theorem, is that the zeroes of the polynomials formed from eigenvectors with unique maximum/minimum eigenvalues of a positive definite Toeplitz matrix are distinct and lie on the unit circle. The proof is based on simple matrix-theoretical results and offers an insight that is lacking in existing proofs. A representation form for Toeplitz matrices is obtained and this form is used for justification of Pisarenko's spectral analysis based on eigenvectors/eigenvalues of the covariance matrix. The theoretical groundwork developed for the proof of the eigenvector property is then used to compare Pisarenko's null eigenvector technique with Burg's maximum entropy analysis, and arguments are presented in favor of the former technique.