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. 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.
