I know that for each inner product $\langle , \rangle_{A}$ on $\mathbb{R}^n$, there is an associated positive definite symmetric matrix $A$ so that $\langle x,y \rangle = x^{T}Ay$. I was wondering if it is possible to find a function relating the angles between $x,y$ in $\mathbb{R}^n$ with the dot product, to the angle in another inner product space.
For example a function $f(\theta,A)$, so that if $\theta$ is the dot product between $x$ and $y$ in $\mathbb{R}^n$, then $f(\theta,A) = \langle x,y \rangle_A$
It is not generally possible to do so. For example, consider the inner product given by $$ A = \pmatrix{ 2&1&0\\ 1&2&0\\ 0&0&1 } $$ Consider $x = (1,0,0)$ and $y = (0,1,0)$. We find that $\langle x,y \rangle_A = x^TAy = 1$, so that $f(0,A) = 1$.
On the other hand, if $x = (1,0,0)$ and $y = (0,0,1)$, we find that $\langle x,y \rangle = x^TAy = 0$, so that $f(0,A) = 0$.