Multiply a vector $\vec x$ by matrix $A$ from the left. $|\vec x|$ is the length of $\vec x$. $|A\vec x|$ is the length of $A\vec x$.
I try to simplify $\displaystyle \frac{|Ax|}{|x|}$, but failed. I can only make it to $\displaystyle \frac{\sqrt{x^TA^2x}}{\sqrt{x^Tx}}$.
If it can't be simplified when $A$ is a general matrix, what about it's symmetric?