Approximating real symmetric Toeplitz matrices using the nearest circulant

TL;DR

This paper derives the nearest symmetric circulant matrix approximation for real Toeplitz matrices in the Frobenius norm, demonstrating improved accuracy over classic methods, especially for matrices with exponential decay.

Contribution

It introduces a closed-form solution for the nearest symmetric circulant approximation of real Toeplitz matrices and compares its performance to traditional methods.

Findings

Nearest circulant approximation outperforms classic methods in finite cases.

Symmetric circulant matrices are uniquely characterized by real eigenvalues.

The approximation's error is explicitly evaluated for matrices with exponential decay.

Abstract

The nearest circulant approximation of a real Toeplitz matrix in the Frobenius norm is derived. This matrix is symmetric. It is proven that symmetric circulant matrices are the only real circulant matrices with all real eigenvalues. The Frobenius norm of the difference between this approximation and the Toeplitz matrix for the case of a Toeplitz matrix displaying exponential decay is evaluated using an expression of $\sum_{n = 1}^N n^k p^n$ in terms of the first $k$ geometric moments. Compared to a classic approximation the nearest circulant displays dramatically better behaviour in any finite cases, though both share the same leading term for large $M$.