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