Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Column bounded matrices and Grothendieck's inequalities

Erik Christensen

TL;DR

The paper develops a factorization framework for complex column-bounded matrices to derive Grothendieck's inequalities in finite dimensions. It proves a central factorization theorem X = C Δ(Λ) Z with l ≤ n^2, a bound on ||C||∞, and entries of Z on the unit circle, enabling a direct route to both Grothendieck inequalities. The results identify the optimal complex constant k_G^C = 4/pi and establish a bound K_G^C ≤ k_G^C/(2 - k_G^C) with explicit numerical windows, while also providing weaker variants and density extension theorems that broaden applicability. Overall, the work links column-norm factorization to operator-space dualities and quantum-correlation interpretations, offering a practical tool for deriving Grothendieck-type bounds from structured matrix factorizations.

Abstract

It follows from Grothendieck's little inequality that to any complex (m x n) matrix X of column norm at most 1, and an 0 <e <1, there exist a natural number q, an (m x q) matrix C with $(1-e)^2 \leq CC^* \leq (4/π) (1 + e)^2$ and an (q x n ) matrix Z with entries in the complex torus such that X= q$^{-(1/2)}$(CZ). Both of Grothendieck's complex inequalities follow from this factorization result.

Column bounded matrices and Grothendieck's inequalities

TL;DR

The paper develops a factorization framework for complex column-bounded matrices to derive Grothendieck's inequalities in finite dimensions. It proves a central factorization theorem X = C Δ(Λ) Z with l ≤ n^2, a bound on ||C||∞, and entries of Z on the unit circle, enabling a direct route to both Grothendieck inequalities. The results identify the optimal complex constant k_G^C = 4/pi and establish a bound K_G^C ≤ k_G^C/(2 - k_G^C) with explicit numerical windows, while also providing weaker variants and density extension theorems that broaden applicability. Overall, the work links column-norm factorization to operator-space dualities and quantum-correlation interpretations, offering a practical tool for deriving Grothendieck-type bounds from structured matrix factorizations.

Abstract

It follows from Grothendieck's little inequality that to any complex (m x n) matrix X of column norm at most 1, and an 0 <e <1, there exist a natural number q, an (m x q) matrix C with and an (q x n ) matrix Z with entries in the complex torus such that X= q(CZ). Both of Grothendieck's complex inequalities follow from this factorization result.
Paper Structure (3 sections, 6 theorems, 42 equations)

This paper contains 3 sections, 6 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Factorizations of column bounded matrices
  3. Proofs of Grothendieck's inequalities based on Theorem \ref{['ColBdFac']}

Key Result

Proposition 1.2

The constant $k_G^{\mathbb C} = 4/\pi$ is the smallest possible posi-tive real, such that for any natural number $n$ and any matrix $D$ in ${\mathcal{D}}_n$ there exists a matrix $R$ in ${\mathcal{R}}_n$ with $D \leq k_G^{\mathbb C} R.$

Theorems & Definitions (13)

  • Definition 1.1
  • Proposition 1.2
  • proof
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • ...and 3 more