I am a practising engineer (who never took real analysis in grad school), and just now, I came across this definition in an applied math textbook (I have paraphrased to simplify the exact wording).
The matrix norm induced from a vector norm can be defined as $$ ||A|| = \underset{x \ne 0}{\text{sup}}\frac{||A x||}{||x||} = \underset{x = 1}{\text{sup}}\ ||A x|| $$
I don't understand the second part of the equality (I had to even look up on YouTube what supremum even means), but I wish to understand this better. Can someone please offer a simple clarification on this?
Sup is like max, but more general. You note $\frac{\|Ax\|}{\|x\|} = \|A\frac{x}{\|x\|}\|$ and $\|\frac{x}{\|x\|}\| = 1$, so $\frac{\|Ax\|}{\|x\|} \leq \sup_{\|v\| = 1}\|Av\|$ for all $x \neq 0$. Hence $\sup_{x \neq 0}\frac{\|Ax\|}{\|x\|} \leq \sup_{\|v\| = 1}\|Av\|$. The other inequality is done similarly.