For two $n \times n$ nonnegative definite Hermitian matrices $A$ and $B$ over a generalthe real number field $F$$\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?
Question1:Is there always a nonsingular matrix $P$ overIf the same field $F$answer is yes, in which makes $P^{H}AP$ and $P^{H}BP$ both diagonalpart 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?
Question2:What special property does this kind of $P$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?
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.