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Assume that $A$ is a symmetric positive definite matrix. For any vectors $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^n$, if the inner product $x^T y\geq0$, then $x^T Ay\geq0$.

I guess the assertion is correct but I have no idea how to prove it. Thank you.

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No. Take the case where $x = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$, $ y =\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ and $A = \begin{bmatrix} a & c \\ c & b \end{bmatrix}$ being some positive definite matrix.

Then, $x^T y = 0$. But, $x^T A y = c$. Now, you can easily construct a p.d. matrix such that $c < 0$ (for example, $a=1, b = 1, c = -1/10$).

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  • $\begingroup$ Thank you Batman. I just asked a stupid question... $\endgroup$ – Ning Zheng Apr 8 '16 at 8:51

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