# Thread: Derivative Problem

1. ## Derivative Problem

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

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

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

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

3. Following up on this problem, it says what I've just proven and the implicit function theorem imply that $\displaystyle G$ 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 $\displaystyle W$ ?

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