For a matrix $A$, let $\|A\|_2$ denote the spectral norm (operator norm with euclidean norm used in both spaces). And for a vector $x$, let $\|x\|_2$ denote the euclidean norm.
Is the following inequality correct?
$ \|Ax\|_2 \leq \|A\|_2 \|x\|_2 $
using the first property listed here.
I have seen from a previous question, they have the inequality
$ \|Ax\|_2 \leq \|A\|_F \|x\|_2 $
which is not as good in general as from this $\|A\|_2 \leq \|A\|_F \leq \sqrt{r}\|A\|_2$