2024/05/10 by Bandeira, Afonso S., Mixon, Dustin G., Steinerberger, Stefan
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2405.06154
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 ∑k xk xk^*=A ⇒ ∑k ‖xk‖21 ≥ c √(n) ‖A‖1, where c is independent of n. This provides a lower bound for the Balan--Jiang matrix problem. The construction is probabilistic.