Is there a name for this type of vector norm?

Question

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.

Answer 1

That's only a norm if $A$ is positive definite. If $A$ is positive definite, then I would call what you describe "the $A$ norm".

Answer 2

I think the best you can say is that $||\cdot||_A$ is a norm induced by an inner product.

(Not all norms are like that.)

The matrix $A$ is the Gram matrix of that inner product with respect to the canonical basis of $\mathbb R^n$ (provided $A$ is positive definite).

Answer 3

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

$$ B(x,x)=<x,x>_B=x^tBx $$