Circulant matrices: norm, powers, and positivity

TL;DR

This paper investigates conditions under which the spectral norm of real and complex circulant matrices equals the modulus of their row or column sum, linking this to positivity of matrix powers and extending previous results.

Contribution

It improves existing conditions to necessary and sufficient ones for the spectral norm of circulant matrices and generalizes findings to complex matrices.

Findings

Established necessary and sufficient conditions for spectral norm equality

Connected spectral norm properties to positivity of matrix powers

Generalized results to complex circulant matrices

Abstract

In their recent paper "The spectral norm of a Horadam circulant matrix", Merikoski, Haukkanen, Mattila and Tossavainen study under which conditions the spectral norm of a general real circulant matrix ${\bf C}$ equals the modulus of its row/column sum. We improve on their sufficient condition until we have a necessary one. Our results connect the above problem to positivity of sufficiently high powers of the matrix ${\bf C^\top C}$. We then generalize the result to complex circulant matrices.