What can we say about, $b=\langle x^TA,x \rangle$?

Question

I wish to comment if the quantity, defined by, \begin{equation} b=\langle x^T A,x \rangle \end{equation} is either positive or negative? Here in this expression $x \in \mathbb{R}^{n}$ and $A \in \mathbb{R}^{n\times n}$ is a rank deficient matrix. I want to know if $b \geq 0$? Please help me prove it! Thanks for your time!

Answer 1

No reason for $x^TAx$ to be positive. Consider $$A = \begin{bmatrix} 0 & 0 \\ 0 & -1 \end{bmatrix}, x = \begin{bmatrix} 0 \\ 1 \end{bmatrix} $$

Then $x^TAx = -1$.