Let $A$ and $B$ are positive definite and positive semi-definite matrices, respectively. $A – B$ is positive semi-definite.
Is it true that $(I \circ A – I \circ B)$ is positive-semidefinite?
I believe this statement is true. Because
$$
(I \circ A – I \circ B) = I \circ (A – B)
$$
and the Hadamard product of two positive (semi)-definite matrices is also positive (semi)-definite. Is it a valid argument?
I don't think $(C \circ A – C \circ B)$ is a positive semidefinite matrix for any arbitrary positive definite matrix $C$.
Best Answer
The question is equivalent to whether or not $I\circ A$ is positive semi definite if $A$ is so. But $I\circ A$ is nothing but the diagonal elements of $A$. If $A≥0$, denote with $A_{ij}$ the components of $A$ and $e_i$ the standard basis of $\Bbb C^n$, then $$A_{ii}= \langle e_i , A e_i\rangle ≥0$$ by positivity and every diagonal element of the diagonal is $≥0$. So $I\circ A$ is diagonal with all entries $≥0$, hence also positive semi-definite.