# Math Help - Derivative Problem

1. ## Derivative Problem

Let $E$ be the set of all 2n x 2n matrices of the form $C=\left( {\begin{array}{cc}
A & -B \\
B & A \\
\end{array} } \right)$
, 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.

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

3. 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?

4. Did you find the dimension of $W$ ?