The p-norm of circulant matrices via Fourier analysis

Abstract

Abstract A recent work derived expressions for the induced p-norm of a special class of circulant matrices A(n, a, b) ∈ ℝn×n, with the diagonal entries equal to a ∈ ℝ and the off-diagonal entries equal to b ≥ 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. The results comprise an exact expression for ǁAǁp, 1 ≤ p ≤ ∞, where A = A(n, a, b), a ≥ 0 and for ǁAǁ2 where A = A(n, −a, b), a ≥ 0; for the other p-norms of A(n, −a, b), 2 < p < ∞, upper and lower bounds are derived.