The p-norm of some matrices

TL;DR

This paper calculates the operator p-norms of specific matrices, introduces logarithmic affine matrices, and provides exact p-norm values for these matrices, including a magic square example.

Contribution

It defines logarithmic affine matrices and computes their p-norms exactly, including a novel example involving a magic square matrix.

Findings

The p-norm of the magic square matrix is 15 for all p.

Logarithmic affine matrices have computable p-norms.

Exact p-norms are derived for a class of matrices.

Abstract

We compute the operator $p$-norm of some $n\times n$ complex matrices, which can be seen as bounded linear operators on the $n$ dimensional Banach space $\ell^p(n)$. The notion of logarithmic affine matrices is defined, and for such a matrix its $p$-norm is computed exactly. In particular, a matrix $A=\begin{pmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{pmatrix}$ which corresponds to a magic square belongs to the class of logarithmic affine matrices, and its $p$-norm is equal to $15$ for any $p\in [1,\infty]$.