Upper bound of $\frac{\|AQx\|_2}{\|Ax\|_2}$ with $Ax\neq 0,\forall x$?

Question

I am studying matrix norm right now, based on the definition of norm on matrix, we know that $\|A\|_2 = \sup_{x} \frac{\|AQx\|_2}{\|Qx\|_2}$ for some given matrix $Q$. Hence the upper bound for $\frac{\|AQx\|_2}{\|Qx\|_2}$ is $\|A\|_2$. Right now I got stuck in finding the upper bound for $\frac{\|AQx\|_2}{\|Ax\|_2}$.

Thank you Kavi for pointing out one special case in which $Ax=0$. If we are restricted ourself on the matrix such that there does not exist $x$ such that $Ax=0$. For example, we can focus ourself on invertible square matrix.

Answer 1

$$\sup_{Ax \neq 0}\frac{\|AQx\|_2}{\|Ax\|_2}=\sup_{y\neq 0} \frac{\|AQA^{-1}y\|_2}{\|y\|_2}=\|AQA^{-1}\|_2$$

Answer 2

There is no such bound. If $||AQx||_2 \leq C||Ax||_2$ then $AQx=0$ whenever $Ax=0$, but this is not true in general.