For two $n \times n$ nonnegative definite Hermitian matrices $A$ and $B$ over the real number field $\mathbb R$:

Question1:Is there always a nonsingular matrix $P$ over the same field $F$ which makes $P^{H}AP$ and $P^{H}BP$ both diagonal?

If the answer is yes, in which part of the matrix theory(or L.A.) can I find such results? If the answer is no, whether there's a similar result or some counterexample?

Question2:What special property does this kind of $P$ have?

Will it still be Hermitian or have some rank inequality in relation with the $rank(A)$ and $rank(B)$, like this sort.

Question3:What is its corresponding background in Abstract algebra?

For example, the Jordan decomposition correspond to the primary decomposition, like this sort.

Moreover I want to know if this result has any application?

Most probably in Lie algebra, because this proposition is taken from some Lie algebra course.