The $p$-norm of circulant matrices

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, $1\le p\le \infty$. 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 $p=2$.

14 pages, to be published in Linear and Multilinear Algebra

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