Say we have two coordinate systems, $A$ and $B$, a matrix $M$ that acts on vectors from coordinate system $A$, and $C$ a matrix that takes vectors from $B$ to $A$.
Let's say that $M$ is unitary (meaning that it satisfies $M^{-1} = M^*$.
To my knowledge, $C^{-1}MC$ will be a matrix that acts on vectors in $B$ in the same way that they would act on $A$. Will this matrix still be unitary?
If the basis vectors in $A$ and $B$ are both orthonormal, my intuition says that it should be true; but I don't know how to prove this. Is it even true?
$(C^{-1}MC)^{-1} = C^{-1}M^{-1}C = C^{-1}M^{*}C $
$(C^{-1}MC)^{*} = C^{*}M^{*}(C^{-1})^{*}$
They are equal if $C^{-1}=C^{*}$.
If $C$ is real and orthonormal, then the above is true.
If $C$ is complex, then $C^{-1}=C^{*}$ may not be true and hence the equality would break.