Is $\sup_{x \in S^{n-1}} \vert \langle Ax,x\rangle\vert$ equal to $\sup_{x,y \in S^{n-1}} \langle Ax,y\rangle$ if $A$ is symmetric

Question

Problem: Let $A = (a_{ij})$ be an $n\times n$ matrix. The spectral norm of $A$ is defined as $$\Vert A \Vert = \sup_{x,y \in S^{n-1}} \langle Ax,y\rangle.$$

If $A$ is symmetric, I wonder this norm can be rewritten as $$\Vert A \Vert = \sup_{x \in S^{n-1}} \vert \langle Ax,x\rangle\vert.$$