# Understanding Matrix Norms [duplicate]

I'm trying to gain intuition for matrix norms. I'd like to know why if $A\in\mathbb{R}^{n\times n}$ that $||A||$ is equal to both $\max _{x\in\mathbb{R}^n}\frac{||Ax||}{||x||}$ and $\max_{||x||=1}||Ax||$. Why are these two definitions equivalent?

Observe for any $x \neq 0$, we have that \begin{align} \frac{\|Ax\|}{\|x\|} = \left\| A\frac{x}{\|x\|}\right\| = \|A v\| \end{align} where $v$ is a unit vector.
• Because $\| \, \cdot \, \|$ is a norm and $(1/\| x\|)$ is a scalar. – LucasSilva Feb 10 '18 at 22:37
• Is this definition also equivalent to $||A||=\max_{||x||<1}\frac{||Ax||}{||x||}$? – confusedmath Feb 10 '18 at 22:38
• So basically by dividing the numerator and denominator by $||x||$? – confusedmath Feb 10 '18 at 22:45