Weighted Norms
Definition
Let $W$ be a positive semidefinite $n$ by $n$
matrix.
Then $g: \R ^n \to \R $ defined by $g(x) =
\sqrt{x^\top W x}$ is a norm on $\R ^n$.
Notation
Let $W \in \R ^{n \times n}$, positive semidefinite. Then we denote the norm corresponding to $W$ by $\norm{\cdot }_{W}$. So then, the norm of a vector $x \in \R ^n$ is $\norm{x}_{w}$. Notice that $\norm{x}_{W} = \norm{W^{1/2}x}_2$.
Visualization1
We can compare the Euclidean norm on $\R ^2$
with the weighted norm given by
\[
W = \pmat{
2 & 1 \\
1 & 4 \\
}
\]
- Future editions will visualize these norms as function on $\R ^2$, via contour plots. ↩︎