 Schur product theorem

The entrywise product of two positive semidefinite matrices is positive semidefinite. Writing each matrix as a sum of rank-one Gram matrices reduces the claim to $(u{u}^T)\circ (v{v}^T)=(u\circ v)(u\circ v{)}^T$.

The Schur product theorem is a result in linear algebra related to matrices and their positive semi-definiteness. It establishes a relationship between the Schur product (or Hadamard product) of two matrices and the positive semi-definiteness of those matrices.