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. 



Diagonalize the simultaneous matrices and its background
You might be interested to look at Section 20 on Simultaneous diagonalization of a pair of Hermitian forms of the following book:



> 
> **Prasolov, V. V.
>  [Problems and theorems in linear algebra.](http://staff.math.su.se/mleites/books/prasolov-1994-problems.pdf) 
>  Translations of Mathematical Monographs, 134. American Mathematical Society, Providence, RI, 1994.**
> 
> 
> 


Among other things the following results are proved:
(diagonalizable is in the sense of hermitian).


Consider $n\times n$ hermitian matrices $A$ and $B$.


(1) If $A$ is a positive definite then $A$ and $B$ are simultaneously diagonalizable.


(2) If $A$ is invertible then $A$ and $B$ are simultaneously diagonalizable if and only if $A^{-1}B$ is diagonalizable and all its eigenvalues are real.


(3) If $A$ and $B$ are both nonpositive or nonnegative then $A$ and $B$ are simultaneously diagonalizable. (It seems that, this is the result you are looking for).


(4) If $A$ and $B$ are not simultaneously isotropic then $A$ and $B$ are simultaneously diagonalizable. 


