How do i prove whether a vector is a symmetric matrix?

Question

How do I prove $ \langle Ax, x \rangle \geq 0 $ if $ A $ is a symmetric matrix? Here $ \langle \cdot, \cdot \rangle $ denotes the dot product.

Answer 1

Counterexample:

$$A=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\;x=\begin{pmatrix}0\\1\end{pmatrix}\implies \langle Ax,x\rangle =\langle -x,x\rangle=-1<0.$$

Answer 2

Use the Sylvester's criterion. https://en.wikipedia.org/wiki/Sylvester%27s_criterion.