Column bounded matrices and Grothendieck's inequalities

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 $(1-e{)}^2\le C{C}^{\ast }\le (4/\ddot{I}€)(1+e{)}^2$ 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.