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