Given $x^TAy$, where $x,y\in \mathbb{R}^{n\times 1}$ and $A\in \mathbb{R}^{n\times n}$ is a symmetric positive definite matrix. Does the inequality below hold?
$$\vert x^TAy \vert \le \Vert x\Vert_F\Vert y\Vert_F \Vert A\Vert_2$$
where $\Vert \cdot \Vert_F:=\sqrt{\mbox{trace}(A^TA)}$ is the Frobenius norm and $\Vert \cdot \Vert_2:=\sqrt{\lambda_{\max}(A^TA)}$ is the spectral norm.