Matrix norm definition with constraint

Question

I know that the p-norm of a matrix is defined as:

$\displaystyle \|A\|_p = \max_{x\neq0}\frac{\|Ax\|_p}{\|x\|_p}$

However, how can I conclude the below formula from the above formula?

$\|A\|_p=\max_{\|x\|_p = 1} \|Ax\|_p$

Answer 1

Let $x\neq0$, then $x$ can be written as $x = \alpha u$ where $\|u\|_p = 1$ and $\alpha = \|x\|_p\neq 0$. The matrix norm is then $$ \|A\|_p = \max_{x\neq 0} \frac{\|Ax\|_p}{\|x\|_p} = \max_{\|u\|_p=1} \frac{\|A(\alpha u)\|_p}{\|\alpha u\|_p} = \max_{\|u\|_p=1} \frac{|\alpha|\|Au\|_p}{|\alpha|\|u\|_p} = \max_{\|u\|_p=1} \frac{\|Au\|_p}{\| u\|_p} = \max_{\|u\|_p=1} \|Au\|_p. $$