Do we say a matrix $A$ is positive definite on $\mathbb{R}^n$ or $\mathbb{R}^n \times \mathbb{R}^n$

Question

Suppose $A$ is positive definite, such that for all $x \in \mathbb{R}^n$, we have:

$$x^TAx > 0$$

Do we say $A$ is positive definite on $\mathbb{R}^n$ or $\mathbb{R}^n \times \mathbb{R}^n$

Answer 1

Remember that $x^T$ is the transpose of the vector $x$, so ultimately, the input is just one vector in $\mathbb{R}^n$. Think about the quadratic form as $q(x) = \langle Ax,x\rangle$.

Answer 2

On $\mathbb{R}^n$ makes more sense to me since that's what $A$ is acting on.

Looking at the Google search results for "positive definite on", whenever the context is linear algebra it seem to always mention the vector space next.