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

Abstract

A recent paper [1] computed the induced p-norm of a special class of circulant matrices Apn, a, bq P R nˆn , with the diagonal entries equal to a P R and the off-diagonal entries equal to b ě 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 }A} p , 1 ď p ď 8, where A " Apn, a, bq, a ě 0 and for }A} 2 where A " Apn, ´a, bq, a ě 0; for the other p-norms of Apn, ´a, bq, 2 ă p ă 8, we provide upper and lower bounds.