TITLE: Is there a name for this type of vector norm?
QUESTION [8 upvotes]: In the case of the $\mathcal{l}_2$ norm we have,
$$||\mathbf{x}||_2^2=\mathbf{x}^T\mathbf{x}.$$
I was wondering if there was a type of norm that had a linear operation embedded in it, like this,
$$||\mathbf{x}||^2_A=\mathbf{x}^T A \mathbf{x},$$
where A is a real matrix.

REPLY [2 votes]: What you're looking for is usually associated with a bilinear form, moreover we say:
A scalarproduct on a real vector space $V$ (induced by a bilinear form $B$)  is a symmetric, non-degenerated, positive definite bilinear form. A scalarproduct then induces a norm.
If we are dealing with a finite dimensional real vector space, we can then also write
$$
B(x,x)=<x,x>_B=x^tBx
$$