The $p$-norm of circulant matrices via Fourier analysis

Abstract

A recent paper computed the induced $p$-norm of a special class of circulant matrices $A(n,a,b)\in {\mathbb{R}}^{n\times n}$, with the diagonal entries equal to $a\in \mathbb{R}$ and the off-diagonal entries equal to $b\ge 0$. We provide shorter proofs for all the results therein using Fourier analysis. The key observation is that a circulant matrix is diagonalized by a DFT matrix. We obtain an exact expression for $\Vert A{\Vert }_p,1\le p\le \infty$, where $A=A(n,a,b),a\ge 0$ and for $\Vert A{\Vert }_2$ where $A=A(n,-a,b),a\ge 0$; for the other $p$-norms of $A(n,-a,b)$, $2<p<\infty$, we provide upper and lower bounds.

arXiv comment6 pages; single-column