Combinatorial methods for the spectral p-norm of hypermatrices

TL;DR

This paper extends bounds on the spectral p-norm of hypermatrices using combinatorial methods, improving previous results and establishing new equalities for symmetric nonnegative hypermatrices.

Contribution

It generalizes Schur's bound to r-matrices and introduces combinatorial concepts like r-partite matrices for spectral norm analysis.

Findings

Extended Schur's bound to r-matrices

Proved spectral p-norm equals p-spectral radius for symmetric nonnegative r-matrices when p ≥ r

Provided bounds on spectral p-norm and p-spectral radius

Abstract

The spectral $p$-norm of $r$-matrices generalizes the spectral $2$-norm of $2$-matrices. In 1911 Schur gave an upper bound on the spectral $2$-norm of $2$-matrices, which was extended in 1934 by Hardy, Littlewood, and Polya to $r$-matrices. Recently, Kolotilina, and independently the author, strengthened Schur's bound for $2$-matrices. The main result of this paper extends the latter result to $r$-matrices, thereby improving the result of Hardy, Littlewood, and Polya. The proof is based on combinatorial concepts like $r$-partite $r$-matrix and symmetrant of a matrix, which appear to be instrumental in the study of the spectral $p$-norm in general. Thus, another application shows that the spectral $p$-norm and the $p$-spectral radius of a symmetric nonnegative $r$-matrix are equal whenever $p\geq r$. This result contributes to a classical area of analysis, initiated by Mazur and Orlicz around 1930. Additionally, a number of bounds are given on the $p$-spectral radius and the spectral $p$-norm of $r$-matrices and $r$-graphs.