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$H + H^T$ is a positive definite matrix and $P$ is also a positive definite matrix.

Will $Q = PH + H^TP$ be a positive definite matrix?

In my calculations, it is not positive definite. But I read a paper saying that $Q$ should be positive definite. Is it so?

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    @Qiachu: I am extremely sorry if I posted my question in wrong place. I am not familiar of this system. This is my second question.2011-08-09

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You refer to your calculations; does that mean you already have a counterexample? My calculations seem to agree with yours, as seen in the example $H=\begin{bmatrix}1&0\\1&1\end{bmatrix}$ and $P=\begin{bmatrix}1&0\\0&5\end{bmatrix}$.

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    Fatima, thanks for giving the name of the article, but I do not know precisely what is claimed in the article, and I do not have access to it at the moment. Perhaps there are additional hypotheses there. If you would like to ask further questions on the particular claim made in the article, it would help to provide a little more context, and perhaps an excerpt for those like me who cannot view it.2011-08-09
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If

$\mathbf H=\begin{pmatrix}15&9&7\cr-1&9&-8\cr-3&-9&11\cr\end{pmatrix}$

and

$\mathbf P=\begin{pmatrix}81&-5&30\cr-5&75&-54\cr30&-54&54\cr\end{pmatrix}$

then $\mathbf Q$ isn't positive definite, having two positive and one negative eigenvalues. It should be easy to generate other counterexamples...

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    Oh =) Okay cool2011-08-11