I've been studying linear algebra in my free time for almost a year. I recently watched the series on inner products by MathTheBeautiful on YouTube, which was incredible. Pavel (the lecturer) drives home the point that orthogonality exists only with respect to an inner product. Steve Brunton mentioned in another video that if you multiply 2 vectors $x$ and $y$ by an orthogonal/unitary matrix $Q$, their inner product doesn't change. This makes sense to me intuitively, especially for the standard inner product.
$\langle Qx,Qy\rangle = x^TQ^TQy=x^Ty=\langle x,y\rangle$
Now I'm trying to figure out how that works for the inner product $\langle x,y\rangle_S = x^TSy$, where $S$ is PD. I'm getting confused because I think the matrix $Q$ has to be orthogonal with respect to the inner product $\langle\cdot,\cdot\rangle_S$. That is:
$\langle q_i,q_i\rangle_S = q_i^TSq_i=1$ and $\langle q_i,q_j\rangle_S = q_i^TSq_j=0$
Which would mean $Q^TSQ=I$
But then
$\langle Qx,Qy\rangle_S = x^TQ^TSQy=x^Ty=\langle x,y\rangle$
So the inner product is not preserved anymore, because it's a different inner product. To clarify, what I would expect to end up with, if $Q$ is orthogonal with respect to the inner product $\langle\cdot,\cdot\rangle_S$, is $\langle Qx,Qy\rangle_S =\langle x,y\rangle_S$
Where is my reasoning going in the wrong direction?
Niels