Derivative Problem

Let $E$ be the set of all 2n x 2n matrices of the form $C=\left( {\begin{array}{cc}<br /> A & -B \\<br /> B & A \\<br /> \end{array} } \right)$ , for $A, B\in Mat(n\times n, \mathbb{R})$ , and let $G=\{C\in E : CC^T=I\}$ . Further, let $f(C)=CC^T$ . This maps $E$ into a subspace $W=\{P\in E : P^T = P\}$ .

Compute the derivative $(Df)_C$ and show it's surjective when $C\in G$ .

So I computed the derivative and got $(Df)_C(U)=CU^T+UC^T$ . For $C\in G$ , this becomes $CU^T+UC^{-1}$ . So for $P\in W$ , somehow I need to come up with a matrix $U\in E$ such that $(Df)_C(U)=P$ , but I'm not seeing it at all.

Put $U := \frac 12 PC$ for $P\inW$ .

Following up on this problem, it says what I've just proven and the implicit function theorem imply that $G$ is a manifold, and to determine its dimension. How do you go about determining the dimension of a manifold?

Did you find the dimension of $W$ ?