[LA] Diagonal two matrices simultaneously

If we have two matrices, $A$ and $B$ with $A$ being positive definite and $B$ being positive semidefinite matrices.

Firstly, from the eigen decomposition of $A$, we know that there exists an orthogonal matrix $V_A$ such that

$$V_A^T A V_A = I$$

Then from the symmetricity property of $B$, we know $V_A^T B V_A$ is also symmetric. So there exists an orthogonal matrix $V_{V_A B}$ such that

$$V_{V_A B}^T V_A^T B V_{A} V_{V_A B} = \Theta$$
where $\Theta$ is a diagonal matrix.

So, if we let $V = V_A V_{V_A B}$, we have

$$\begin{align} V^T A V &= I\\ V^T B V &= \Theta \end{align}$$