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I am reading on my own the Lectures on the Geometry of Manifolds (http://nd.edu/~lnicolae/Lectures.pdf ) , and got stuck in solving the exercise 1.1.3 (b) .

The 1.1.3 (b) is :

Let F: $U\rightarrow U$ be defined as $A\rightarrow A^{-1}$. Show that $% D_{A}F(H)=-A^{-1}HA^{-1}$ for any $n\times n$ matrix $H.$

$D_{A}F(H)$ is the Frechet derivative of F at A. H I guess should be the small "drifting", so that it is actually trying to calculate the derivative of $(A+tH)^{-1}$

I tried to expand $(A+H)^{-1}$ as $(I+A^{-1}H)^{-1}A^{-1}=I-A^{-1}H+\frac{% A^{-2}H^{2}}{2!}-\frac{A^{-3}H^{3}}{3!}+...,$ but it doesn't looks like $% -A^{-1}HA^{-1}$

Some one can give me a hint?

1 Answers 1

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You are almost correct, you made a mistake in your expansion:

Assuming $\|A^{-1}H\| < 1$, we have

$(A+H)^{-1} = (A(I+A^{-1}H))^{-1} = (I+A^{-1}H)^{-1}A^{-1} = \sum_{k=0}^\infty (-1)^k(A^{-1}H)^k A^{-1}$. This gives $(A+H)^{-1} = A^{-1}-A^{-1} H A^{-1} + o(H)$, from which it follows that $ D F(A)(H) = -A^{-1} H A^{-1}$.

(It doesn't matter here, but you can't assume that $A^{-1}$ and $H$ commute. And you have factorials in your expansion, which are incorrect.)