Let $W$ be an invertible matrix. Show that the map, $\|x\|_W = \|Wx\|_2$ is a norm on $\mathbb{R}^m$.

Question

There are two things which I'm unsure about with this question which I believe are stopping me from proceeding to find a solution.

1) Since we're told that the map given is a valid norm in $\mathbb{R}^m$ and that it is invertible we know that $W$ must be a $m \times m$ matrix right?

2) What does the norm with a matrix as a subscript mean? Does it mean the induced norm given by the dimensions of a matrix (E.g. A is $2 \times 3$ so $\|x\|_A = \|x\|_{2,3}$) or something else?

Answer 1

If $W$ is injective then $\|x\|_W = \|Wx\|_2$ is a norm.

$\|tx\|_W = \|W (tx)\|_2 = |t| \|Wx\|_2 = |t| \|x\|_W$.

$\|x+y\|_W = \|W(x+y)\|_2 \le \|Wx\|_2 + \|Wy\|_2 = \|x\|_W + \|y\|_W$.

Suppose $\|x\|_W = 0$, then $\|Wx\|_2 = 0$ and hence $Wx = 0$. Since $W$ is presume injective we have $x=0$.

Note, it does not have to be the Euclidean norm $\| \cdot \|_2$, any norm will do.