Column bounded matrices and Grothendieck's inequalities

TL;DR

This paper presents a new factorization approach for complex matrices with bounded column norms, deriving Grothendieck's inequalities through this method, which offers a deeper understanding of matrix factorizations and inequalities.

Contribution

It introduces a novel matrix factorization technique that directly leads to Grothendieck's inequalities for complex matrices, expanding the theoretical framework.

Findings

Established a factorization for matrices with bounded column norms.

Derived Grothendieck's inequalities from the new factorization.

Provided bounds on the matrices involved in the factorization.

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 $(1-e)^2 \leq CC^* \leq (4/\pi) (1 + e)^2$ 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.