I am having trouble understanding what the following question is asking:
For the space of real n by n matrices:
Find the Frechet derivatives of A maps to A^2, and A maps to A^-2.
Anyone have any pointers?
Let $\displaystyle f:M_{n \times n} (\mathbb{R}) \longrightarrow M_{n \times n} (\mathbb{R})$ such that $\displaystyle A \mapsto A^2$ then, (I'm assuming you give this space the operator norm):
$\displaystyle \frac{ \Vert f(A+H) - f(A) - Df_A(H) \Vert }{ \Vert H \Vert} = \frac{ \Vert AH + HA - Df_A(H) \Vert }{ \Vert H \Vert }$ and we want this to tend to zero as $\displaystyle H$ tends to zero, which is clearly satisfied if we define $\displaystyle Df_A(H):= AH+HA$, this is clearly linear, and $\displaystyle \Vert AH+HA \Vert \leq \Vert HA \Vert + \Vert AH \Vert \leq \Vert H \Vert \Vert A \Vert + \Vert A \Vert \Vert H \Vert = 2 \Vert A \Vert \Vert H \Vert$ from which it follows that $\displaystyle Df_A(H)$ is a bounded linear operator, and as such is the derivative of $\displaystyle f$ at $\displaystyle A$ doing this for all $\displaystyle A$ we have a map: $\displaystyle Df_{\cdot }: M_{n \times n} (\mathbb{R}) \longrightarrow \mathcal{L} ( M_{n \times n} (\mathbb{R}), M_{n \times n} (\mathbb{R}) )$ where $\displaystyle A \mapsto Df_A(H)$.
The second one is a little trickier since the function is only defined in the subset of nonsingular matrices, and you would have to prove first that this subset is open.