Suppose $A = (a_{ij})$ is positve semidefinite and $B = (b_{ij})$ is negative semidefinite, then is $C = (a_{ij}b_{ij})$ (elementwise product) negative semidefinite?
What I'm concerned is a weaker conclusion: is $\sum_{i,j} a_{ij}b_{ij} \leq 0$?
[Math] Is the elementwise product of a positive semidefinite matrix and a negative semidefinite matrix negative semidefinite
linear algebramatrices
Best Answer
Yes, that's a limiting case of Schur product theorem (and the theorem itself is just an easy consequence of the fact that the Hadamard product of two matrices is just a submatrix of their Kronecker product).
Since $A\circ B$ is negative semidefinite, $\mathbf1^T(A\circ B)\mathbf1\le0$.