# Math Help - Vector Norms

1. ## Vector Norms

Let S be a real and nonsingular matrix, and let ||.|| be any norm on R^n. Define ||.||' by ||x||' = ||Sx||. Show that ||.||' is also a norm on R^n.

I know ||S|| = max ||Sx|| from ||x||=1 but I'm not sure if that applies.

2. You just need to check the axioms for a norm. Let $x,y\in \mathbb R^n$, and $\lambda \in \mathbb R$. Then

Positive definiteness:

$\|x\|'=0\Leftrightarrow \| S\cdot x\| =0$
$\Leftrightarrow S\cdot x=0$ (since $\|\cdot\|$ is a norm)
$\Leftrightarrow x=0$ (since $S$ is nonsingular, and thus invertible)

Positive homogeneity:

$\|\lambda\cdot x\|'=\|S\cdot (\lambda\cdot x)\| = \|\lambda \cdot (S\cdot x)\|$
$=|\lambda|\cdot \|S\cdot x\|$ (for $\|\cdot\|$ is a norm)
$=|\lambda|\cdot \|x\|'$

Triangle inequality:

$\|x+y\|'=\|S\cdot (x+y)\|=\| S\cdot x+S\cdot y\|$
$\leq \|S\cdot x\|+\|S\cdot y\|$ (again, $\|\cdot\|$ is a norm)
$=\|x\|'+\|y\|'$