Abstract
We prove the existence of a positive semidefinite matrix A∈Rn×n such that any decomposition into rank-1 matrices has to have factors with a large ℓ1−norm, more precisely∑kxkxk⁎=A⇒∑k‖xk‖12≥cn‖A‖1, where c is independent of n. This provides a lower bound for the Balan–Jiang matrix problem. The construction is probabilistic.
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