Question
Write $(2n\times 2n)$-matrices in block form
$A=$
$\begin{bmatrix}a&b\\c&d\end{bmatrix}$ $\in Mat_{\mathbb{R}}(2n)$
where each of $a,b,c,d$ is an $n\times n$ block. Define
$J=$
$\begin{bmatrix}0&1\\-1&0\end{bmatrix}$
where 1 is a shorthand for an $n\times n$ identity matrix. Let
$Sp(2n)= \{ A\in Mat_\mathbb{R}(2n)| AJA^T=J \}$,
so that $Sp(2n)=F^{-1}J$, where $F: Mat_{\mathbb{R}}(2n)\rightarrow Skew_{\mathbb{R}}(2n)$ is the map $F(A)=AJA^T$. Here, the codomain is the set of real $n\times n$ matrices satisfying $C=-C^T$.
Find the differential $D_AF: Mat_{\mathbb{R}}(2n)\rightarrow Skew_{\mathbb{R}}(2n)$
Note I can find the differential of a function. However, I can’t seem to handle this, probably because it involves matrices. Your help would be greatly appreciated.