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arXiv · 2109.09728

The $p$-norm of circulant matrices

Ludovick Bouthat·Apoorva Khare·Javad Mashreghi·Frédéric Morneau-Guérin
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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, $1 \leq p \leq \infty$. 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 $p=2$.

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Ludovick Bouthat, Apoorva Khare, Javad Mashreghi, Frédéric Morneau-Guérin. 2021-09-20. The $p$-norm of circulant matrices. https://doi.org/10.1080/03081087.2021.1983513

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