Finding the Frechet Derivative of a Map

**Relayer** · September 16th 2009, 03:49 PM

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?

**Jose27** · September 17th 2009, 02:10 PM

Let $f:M_{n \times n} (\mathbb{R}) \longrightarrow M_{n \times n} (\mathbb{R})$ such that $A \mapsto A^2$ then, (I'm assuming you give this space the operator norm):

$\frac{ \Vert f(A+H) - f(A) - Df_A(H) \Vert }{ \Vert H \Vert} = \frac{ \Vert AH + HA - Df_A(H) \Vert }{ \Vert H \Vert }$ and we want this to tend to zero as $H$ tends to zero, which is clearly satisfied if we define $Df_A(H):= AH+HA$ , this is clearly linear, and $\Vert AH+HA \Vert \leq \Vert HA \Vert + \Vert AH \Vert \leq \Vert H \Vert \Vert A \Vert + \Vert A \Vert \Vert H \Vert = 2 \Vert A \Vert \Vert H \Vert$ from which it follows that $Df_A(H)$ is a bounded linear operator, and as such is the derivative of $f$ at $A$ doing this for all $A$ we have a map: $Df_{\cdot }: M_{n \times n} (\mathbb{R}) \longrightarrow \mathcal{L} ( M_{n \times n} (\mathbb{R}), M_{n \times n} (\mathbb{R}) )$ where $A \mapsto Df_A(H)$ .

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.

**Rebesques** · September 24th 2009, 01:40 PM

J is right, but if you are looking to actually compute the Frechet derivative, try
$\frac{d}{dt}f(A+tH)\big{\vert}_{t=0}$

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