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arxiv: 1906.10787 · v1 · pith:2XGJZAOUnew · submitted 2019-06-25 · 🧮 math.CO

The p-norm of hypermatrices with symmetries

V. Nikiforov This is my paper

Pith reviewed 2026-05-25 16:05 UTC · model grok-4.3

classification 🧮 math.CO
keywords p-normr-matrixhypermatrixspectral radiussymmetrynonnegativemultilinear formtensor norm
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The pith

If a nonnegative r-matrix is symmetric in two indices, its p-norm is attained by identical vectors in those positions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that for a nonnegative r-matrix with symmetry between two indices, the p-norm maximum is reached when the vectors for those indices are set identical. This directly implies that any symmetric nonnegative r-matrix has its p-spectral radius equal to its p-norm for every p at least 2. The result generalizes the familiar 2-norm of ordinary matrices to higher-order arrays while relating two different size measures through the symmetry. Readers focused on hypermatrix theory would care because the equality removes a distinction that otherwise complicates analysis and computation in the nonnegative symmetric setting.

Core claim

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 ith and the jth 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≥2.

What carries the argument

The p-norm of an r-matrix, achieved as the maximum of the associated multilinear form over unit vectors, with index symmetry forcing attainment at repeated vectors when entries are nonnegative.

If this is right

  • For a fully symmetric nonnegative r-matrix the p-spectral radius equals the p-norm for every p at least 2.
  • The equality between radius and norm holds uniformly for all such p without further restrictions on the matrix order r.
  • The vector repetition property reduces the effective dimension of the optimization problem defining the p-norm under the given symmetry.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The vector repetition may reduce the computational search space when numerically approximating the p-norm of large symmetric nonnegative hypermatrices.
  • The same symmetry argument could be tested for other p-values below 2 or for signed entries under additional sign-pattern constraints, though the paper does not pursue these cases.
  • The equality supplies a direct bridge between spectral-radius bounds and norm bounds that might be applied in extremal problems on hypergraphs represented by such matrices.

Load-bearing premise

The r-matrix must have all nonnegative entries, since without nonnegativity the attainment at identical vectors can fail even with symmetry.

What would settle it

A nonnegative r-matrix symmetric in two indices where the p-norm supremum is strictly larger than the value obtained from any choice of vectors with the two symmetric positions using identical unit vectors.

read the original 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 $p\geq2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that if a nonnegative r-matrix (hypermatrix) is symmetric with respect to two indices j and k, then its p-norm is attained at a collection of vectors in which the vectors for indices j and k are identical. It follows that, for any fully symmetric nonnegative r-matrix, the p-spectral radius equals the p-norm whenever p ≥ 2.

Significance. If the stated implication holds, the result supplies a direct link between the multilinear p-norm and the homogeneous p-spectral radius for symmetric nonnegative hypermatrices, extending classical matrix-norm identities to the tensor setting. The explicit use of nonnegativity to force coincidence of the maximizing vectors is a concrete technical contribution that may be useful in spectral hypergraph theory and multilinear optimization.

minor comments (2)
  1. [Abstract] The abstract refers to an “r-matrix” without a one-sentence reminder of the order-r convention; a brief parenthetical in the first sentence would improve immediate readability.
  2. [Introduction / §1] Notation for the p-norm (presumably the supremum of the absolute multilinear form over unit p-norm vectors) and for the p-spectral radius should be introduced with a displayed equation in §1 or §2 so that the equality statement is unambiguous.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation to accept the manuscript. The referee's summary accurately captures the main result, and we appreciate the recognition of its potential utility in spectral hypergraph theory and multilinear optimization.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a direct theorem: for a nonnegative r-matrix symmetric in indices j and k, the p-norm is attained at vectors with the corresponding pair identical; this immediately implies equality of p-norm and p-spectral radius for fully symmetric nonnegative cases when p≥2. The provided abstract and claim description contain no self-definitional loops, no fitted parameters renamed as predictions, no load-bearing self-citations, and no ansatz smuggled via prior work. The result is presented as a consequence of nonnegativity plus symmetry, with the derivation self-contained as a mathematical proof rather than a reduction to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result is a pure existence and equality statement in multilinear algebra; it rests on standard definitions of the p-norm and symmetry but introduces no fitted parameters or new entities.

axioms (1)
  • standard math Standard definition of the p-norm for r-matrices and the notion of symmetry in specified indices.
    Invoked implicitly in the statement of the theorem in the abstract.

pith-pipeline@v0.9.0 · 5604 in / 1126 out tokens · 35617 ms · 2026-05-25T16:05:02.785376+00:00 · methodology

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Reference graph

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