If $A$ is positive real symmetric matrix and $B$ is real symmetric, does there exist some $V$ such that $V^{T}AV$=$I$ and $V^TBV$ is diagonal matrix?

Question

Let's say we have two matrices $A$ (a positive real symmetric matrix) and $B$ (a real symmetric matrix). And let us suppose that in general $A$ and $B$ don't commute with each other. Then,

Q. Is it possible to have a transformation matrix $V$ such that $V^{T}AV$=$I$ and $V^TBV$ is diagonal? Please provide proof.

I can see that one can find an orthogonal transformation such that it diagonalizes (but doesn't make it unit matrix $I$) $A$ (as it is a symmetric matrix) but it will not always diagonalize $B$. Is it even possible to find such a transformation $V$?

Answer 1

Yes. Let $A = S^TS$ and $R=S^{-1}$. Since $R^TBR$ is real symmetric matrix you can find $U$ is an orthogonal matrix such that $U^TR^TBRU=\Gamma$ where $\Gamma$ is a diagonal matrix?

Let $V = RU$ can you finish the proof?