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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?

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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.

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  • $\begingroup$ Why is the first equality true? $\endgroup$ – confusedmath Feb 10 '18 at 22:31
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    $\begingroup$ Because $\| \, \cdot \, \|$ is a norm and $(1/\| x\|)$ is a scalar. $\endgroup$ – LucasSilva Feb 10 '18 at 22:37
  • $\begingroup$ Is this definition also equivalent to $||A||=\max_{||x||<1}\frac{||Ax||}{||x||}$? $\endgroup$ – confusedmath Feb 10 '18 at 22:38
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    $\begingroup$ By the scaling argument, the definitions are the same. $\endgroup$ – Jacky Chong Feb 10 '18 at 22:42
  • $\begingroup$ So basically by dividing the numerator and denominator by $||x||$? $\endgroup$ – confusedmath Feb 10 '18 at 22:45

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