$A$ is a symmetric matrix, $x$ is a vector. If $(Ax,Ax)=(x,x)$,then $Ax=x$ or $Ax=-x$. Is it true? How to prove it?

Question

$A$ is a symmetric matrix, $x$ is a vector. If $(Ax,Ax)=(x,x)$,then $Ax=x$ or $Ax=-x$.Is it true? How to prove it? or give some examples. Thanks!

Answer 1

This is not true. Consider the matrix $$ A = \begin{bmatrix}1&0\\0&-1\end{bmatrix} $$ We do have $(Ax, Ax) = (x, x)$ for any $x\in \Bbb R^2$, but a vector like $\left[\begin{smallmatrix}1\\1\end{smallmatrix}\right]$ doesn't get sent to itself or its negative.