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?
2026-03-26 15:16:44.1774538204
Understanding Matrix Norms
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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.