Let $ A \in \mathbb R^{n \times n} $ be a positive semidefinite matrix, and let $x, y \in \mathbb R^n $. If $\langle x, y \rangle \geq 0 $ then ...

Question

Can we prove the following statement?

Let $ A \in \mathbb R^{n \times n} $ be a positive semidefinite matrix, and let $x, y \in \mathbb R^n $. If $\langle x, y \rangle \geq 0 $ then $\langle x, Ay \rangle \geq 0 $.

Answer 1

The statement is false. Consider $$ x = \pmatrix{4\\1}, \quad y = \pmatrix{1\\4}, \quad A = \pmatrix{2&-1\\-1&2}. $$ We have $\langle x,y \rangle = 8 > 0$ but $\langle x,Ay \rangle = -1 < 0$.