The condition number of a random banded Toeplitz matrix is typically large

TL;DR

This paper investigates the conditioning of random banded Toeplitz matrices, revealing that symmetry in bandwidth leads to well-conditioned matrices, while asymmetry results in ill-conditioning, emphasizing the impact of structural constraints.

Contribution

It demonstrates that the asymmetry in bandwidth of random banded Toeplitz matrices determines their conditioning, a novel insight into structured random matrices.

Findings

Symmetric bandwidth matrices are well conditioned with high probability.

Asymmetric bandwidth matrices tend to be ill conditioned.

Structural constraints significantly influence the numerical behavior of random matrices.

Abstract

It is well known that square matrices with independent and identically distributed (iid) random entries are typically well conditioned. A natural question is whether this favorable behavior persists for random matrices whose entries obey additional structure, i.e., their position inside of the matrix. A prominent class of structured matrices is given by {\it Toeplitz matrices}, characterized by constant diagonals. A particular tractable subclass is that of circulant matrices, whose additional characteristic (its entries {\it circulate} row by row) allows one to express their conditioning in terms of the localization of the zeros of a associated polynomial. When the entries of a circulant matrix are iid, the matrix is well conditioned precisely when the corresponding random polynomial has no zeros on the unit circle. This connection is especially relevant because, as the degree of a random polynomial increases, its zeros tend to concentrate near the unit circle, making it a delicate problem to quantify how close the closest zeros lie to the unit circle. Another notable family within the Toeplitz class is that of {\it banded Toeplitz matrices}, namely matrices for which only finitely many diagonals around the main diagonal may be nonzero. These matrices have been extensively studied in Operator Theory, nevertheless, despite their apparent simplicity, they raise subtle questions regarding the behavior of their condition numbers. In present work we show that the bandwidth asymmetry plays a decisive role: if the band contains $r$ diagonals below and $s$ diagonal above the main diagonal, then if $r=s$ the banded Toeplitz matriz is well conditioned with high probability, whereas if $r\neq s$ it is typically ill conditioned. This highlights that structural constraints can have a impact on the numerical behavior of random matrices.