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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

1 Answers

One Answer

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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