# The p-norm of hypermatrices with symmetries

## TL;DR

This paper investigates the properties of the p-norm of symmetric nonnegative hypermatrices, showing that symmetry simplifies the computation of spectral radius and norm, with implications for tensor analysis.

## Contribution

It establishes that for symmetric nonnegative hypermatrices, the p-norm is achieved by vectors with certain symmetries, linking spectral radius and p-norm for p≥2.

## Key findings

- p-norm is attained by vectors with identical components for symmetric indices
- Spectral radius equals p-norm for symmetric nonnegative hypermatrices when p≥2
- Symmetry reduces complexity in computing hypermatrix norms and spectral properties

## Abstract

The $p$-norm of $r$-matrices generalizes the $2$-norm of $2$-matrices. It is shown that if a nonnegative $r$-matrix is symmetric with respect to two indices $j$ and $k$, then the $p$-norm is attained for some set of vectors such that the $i$th and the $j$th vectors are identical. It follows that the $p$-spectral radius of a symmetric nonnegative $r$-matrix is equal to its $p$-norm for any $p\geq2$.