Orthogonal transformation stays orthogonal if we change the inner product?

Question

Let $(E,\langle \cdot , \cdot \rangle)$ be a finite euclidian space and let $f$ be an orthogonal linear transformation defined on it. Does $f$ stays orthogonal if we change the underlying inner product of $E$? If this is not true in general, what are the conditions that we can impose on $f$ to make it true?

Answer 1

This is not true in general, because orthogonality is a condition that depends on the inner product choice. Requiring that a map preserves every possible inner product is equivalent to requiring that there is a matrix $B$ such that: $$(Bx)^TA(By)=x^TAy$$ for every positive definite matrix $A$. This implies that: $$B^TAB=A$$ For $A=I_n$ we have that $B^T=B^{-1}$, so: $$B^{-1}AB=A$$ So: $$AB=BA$$ for every positive definite matrix $A$. I think (NOT SURE THOUGH) that this should be enough to say that $B=\lambda I_n$ for some $\lambda$. But since: $$B^TAB=A$$ we have: $$\lambda^2A=A$$ i.e. $\lambda=\pm 1$ and $B=\pm I_n$.