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I need to show that hadamard (schur) product $ A \circ B$ can be positive definite even if not both A and B are positive definite. It would be nice to see a simple example which prooves this.

I know that According to Schur Product Theorem, if both A and B are positive definite, then their hadamard (schur) product $ A \circ B$ is also positive definite.

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    $\begingroup$ We usually write Hadamard's name with a capital H. Schur's also. $\endgroup$ May 1, 2014 at 18:26

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Consider the $1\times1$ matrices $A=(-1)$ and $B=(-1)$.

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  • $\begingroup$ Any example for 2x2 matrix will be appreciated $\endgroup$
    – Almas
    May 1, 2014 at 18:33
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    $\begingroup$ Consider $A=\begin{pmatrix}-1&0\\0&-1\end{pmatrix}$ and $B=\begin{pmatrix}-1&0\\0&-1\end{pmatrix}$. You can construct examples of all sizes, as you see! $\endgroup$ May 1, 2014 at 18:34

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