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Math Help - Finding the Frechet Derivative of a Map

  1. #1
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    Unhappy Finding the Frechet Derivative of a Map

    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?
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  2. #2
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    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.
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  3. #3
    Super Member Rebesques's Avatar
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    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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