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- Apr 14, 2013

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Given that $C \in \mathbb{R}^{n,n}$ is symmetric and positive definite and $D \in \mathbb{R}^{n,n}$.

I have to show that $D^TCD$ is positive definite $\Leftrightarrow $ $D$ is invertible.

For the direction $\Rightarrow $:

$D^TCD$ is positive definite, that means that $\forall x \in \mathbb{R}^n\setminus \{0\} :$ $ x^T D^TCD x >0$.

How can I continue?