Induced Matrix Norm Property Help

Question

I am trying to see how my textbook went from saying:

$\max_{x \neq 0} ||A(x/||x||)|| = \max_{||x||=1} ||Ax||$.

But I am having some trouble seeing this. (Here, $A$ is a matrix, and $x$ is a vector). Thanks.

Answer 1

Note that the quantity $$||A\cdot {x\over ||x||}||$$ is invariant w.r.t. the magnitude of $x$, since for all $k\ne 0$ $$||A\cdot {kx\over ||kx||}||{=||A\cdot {x\over ||x||}||\\=||A\cdot {kx\over k||x||}||\\=||A\cdot {x\over ||x||}||}$$