Appendix for Deterministic Symmetric Positive Semidefinite Matrix Completion

First, note that by assumption rank{A} > 0. Let Ω1 = ρ1 × ρ1 and Ω2 = ρ2 × ρ2 be the two index sets in the theorem. By assumption we have ρ1 × ρ1 ∪ ρ2 × ρ2 = Ω and Ω 6= [n]× [n]. If A1 is not met, then ρ1 ∪ ρ2 6= [n], and from lemma 6 we can conclude recovery of A is impossible. If ρ1 ∪ ρ2 = [n], but A2 is not met then ι2 = |ρ1 ∩ ρ2| < r so it must be that rank{A(ι2, ι2)} < r. Further, by assumption rank{A(ρ1, ρ1)} = rank{A(ρ2 \ ι2, ρ2 \ ι2)} = r. By lemma 6, we can again conclude that matrix recovery is impossible. Finally, if A3 is not met, then again rank{A(ι2, ι2)} < r and we can conclude matrix recovery is impossible.