# Is $(I circ A - I circ B)$ positive semi-definite if $A$, $B$ and $A - B$ are positive semi-definite?

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 $$C$$.

Mathematics Asked on November 14, 2021

The question is equivalent to whether or not $$Icirc A$$ is positive semi definite if $$A$$ is so. But $$Icirc A$$ 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 $$A_{ii}= langle e_i , A e_irangle ≥0$$ by positivity and every diagonal element of the diagonal is $$≥0$$. So $$Icirc A$$ is diagonal with all entries $$≥0$$, hence also positive semi-definite.

Answered by s.harp on November 14, 2021

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