A lower bound for the Balan–Jiang matrix problem

Abstract

We prove the existence of a positive semidefinite matrix $A\in R^{n\times n}$ such that any decomposition into rank-1 matrices has to have factors with a large ${\ell }^1-$norm, more precisely$\sum _k{x}_k{x}_k^{⁎}=A\Rightarrow \sum _k{\Vert x_k\Vert }_1^2\ge c\sqrt{n}{\Vert A\Vert }_1,$ where c is independent of n. This provides a lower bound for the Balan–Jiang matrix problem. The construction is probabilistic.