Is $x^TAx = y^TAy$ if $\|x\| =\|y\|$ I have a symmetric matrix A and the vectors x,y. I'm struggling to show that $x^TAx = y^TAy$ if $\|x\|=\|y\|$ for example taking the 2-norm, could anyone help me out here? Thanks a lot!

No it is not correct, just consider $A=\left[\begin{matrix} 2 & 0 \\\ 0 & 1\end{matrix} \right]$ and the two canonical basis vectors.