Could you please let me know an answer to the following question on simultaneous congruence of two matrices. This question came up while trying to handle a system of PDE.
QUESTION:
Let $A,B\in M_n(\mathbb{R})$, let $A$ be symmetric and let $B$ skew-symmetric. It is known that
i) Sylvester's Theorem: $A$ is congruent to $D$ i.e. for some $Q\in GL_n(\mathbb{R})$, $Q^t A Q=D$, where $$D:=\operatorname*{diag}\left(\underbrace{1,\ldots,1}_{p\text{-times}},\underbrace{-1,\ldots,-1}_{q\text{-times}},\underbrace{0,\ldots,0}_{(n-p-q)\text{-times}}\right),$$ where $p,q$ stand for the number of positive and negative eigenvalues of $A$, respectively.
ii) Darboux Theorem: $B$ is congruent to $J_m$ i.e. for some $R\in GL_n(\mathbb{R})$, $R^t B R=J_m$, where $$J_m:=\operatorname*{diag}\left(\underbrace{\left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \\ \end{array} \right),\ldots,\left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \\ \end{array} \right)}_{m\text{-times}},0,\ldots,0 \right),$$ where $\operatorname*{rank}B=2m$.
When is $A,B$ simultaneously congruent i.e. there exists a $P\in GL_n(\mathbb{R})$ such that
$$P^t A P =D\text{ and }P^t B P = J_m?$$ Is there any necessary and sufficient condition?
