The $p$-norm of circulant matrices

TL;DR

This paper derives explicit formulas and bounds for the induced p-norms of circulant matrices acting on Euclidean space, with exact results for nonnegative entries and specific p-values, enhancing understanding of their operator norms.

Contribution

It provides explicit formulas for the p-norms of circulant matrices with nonnegative entries and bounds for other p-values, including the 2-norm, advancing the analysis of these matrix norms.

Findings

Explicit formula for p-norm when entries are nonnegative.

Exact 2-norm determination.

Optimal bounds at p=1, p=2, and p=∞.

Abstract

In this note we study the induced $p$-norm of circulant matrices $A(n,\pm a, b)$, acting as operators on the Euclidean space $\mathbb{R}^n$. For circulant matrices whose entries are nonnegative real numbers, in particular for $A(n,a,b)$, we provide an explicit formula for the $p$-norm, $1 \leq p \leq \infty$. The calculation for $A(n,-a,b)$ is more complex. The 2-norm is precisely determined. As for the other values of $p$, two different categories of upper and lower bounds are obtained. These bounds are optimal at the end points (i.e. $p=1$ and $p = \infty$) as well as at $p=2$.