On the Asymptotic Equivalence of Circulant and Toeplitz Matrices

TL;DR

This paper demonstrates that Hermitian Toeplitz matrices are asymptotically equivalent to circulant matrices, enabling efficient eigenvalue analysis and approximation in large-scale applications like coding and filtering.

Contribution

It establishes the asymptotic equivalence of Toeplitz and circulant matrices and shows this leads to convergence of eigenvalue estimates, facilitating computational efficiency.

Findings

Eigenvalues of Toeplitz matrices converge to those of circulant matrices asymptotically.

Eigenvalue estimates accurately approximate extremal eigenvalues for broad classes of Toeplitz matrices.

Supports using FFT-based methods for eigenvalue analysis in large systems.

Abstract

Any sequence of uniformly bounded $N\times N$ Hermitian Toeplitz matrices $\{\boldsymbol{H}_N\}$ is asymptotically equivalent to a certain sequence of $N\times N$ circulant matrices $\{\boldsymbol{C}_N\}$ derived from the Toeplitz matrices in the sense that $\left\| \boldsymbol{H}_N - \boldsymbol{C}_N \right\|_F = o(\sqrt{N})$ as $N\rightarrow \infty$. This implies that certain collective behaviors of the eigenvalues of each Toeplitz matrix are reflected in those of the corresponding circulant matrix and supports the utilization of the computationally efficient fast Fourier transform (instead of the Karhunen-Lo\`{e}ve transform) in applications like coding and filtering. In this paper, we study the asymptotic performance of the individual eigenvalue estimates. We show that the asymptotic equivalence of the circulant and Toeplitz matrices implies the individual asymptotic convergence of the eigenvalues for certain types of Toeplitz matrices. We also show that these estimates asymptotically approximate the largest and smallest eigenvalues for more general classes of Toeplitz matrices.