Positive Semidefinite Rank
summary
Definition: smallest K for which KxK PSD matrices A_i, B_j satisfy <A_i,B_j>=M_ij, so positive semidefinite rank measures factorizations through PSD cones. Concrete: 3x3 derangement matrix has usual rank 3 but PSD rank 2. Inequalities relate rank, PSD rank and nonnegative rank and give sqrt(rank) lower bound; square root rank equals the minimum K when all factors are rank one. PSD rank equals minimal K for a projected spectrahedron sandwiching via a slack matrix. Complexity: square root rank is NP‑hard and many lower‑bound questions remain open.
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