On the banded Toeplitz structured distance to symmetric positive semidefiniteness

TL;DR

This paper addresses the challenge of finding the closest positive definite banded Toeplitz matrix to a given banded matrix, reviewing existing theory and proposing a straightforward method for this problem.

Contribution

It introduces a simple approach to compute the nearest banded positive definite Toeplitz matrix, extending prior results and addressing the positive definiteness constraint.

Findings

Provides a practical method for the problem

Extends theoretical understanding of Toeplitz matrices

Offers insights into positive definiteness constraints

Abstract

This paper is concerned with the determination of a close real banded positive definite Toeplitz matrix in the Frobenius norm to a given square real banded matrix. While it is straightforward to determine the closest banded Toeplitz matrix to a given square matrix, the additional requirement of positive definiteness makes the problem difficult. We review available theoretical results and provide a simple approach to determine a banded positive definite Toeplitz matrix.