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I'm trying to work this problem for exam prep:

Define $\Phi:{GL}_n(\mathbb{R})\rightarrow M_n(\mathbb{R})$ by $\Phi(X)=X^{-1}$. Show that $\Phi\in C^{1}({GL}_n(\mathbb{R}))$, and $D\Phi(X)(Y)=-X^{-1}YX^{-1}.$

I know that the inverse map is continuous, so the $C^1$ property should follow immediately from showing that $D\Phi(X)(Y)=-X^{-1}YX^{-1}.$ I cannot seem to show this. At my disposal, I have the standard Frechet derivative definition, as well as the definition of the derivative as the best linear approximation (that is, $f(x+y)=f(y)+Df(x)(y)+R(x,y),$ with $\frac{||R(x,y)||}{||y||}\rightarrow 0$). I have been trying to use the first definition (I heavily prefer it), to no avail. The second definition does not seem to be much help, since I'd have to expand $(X+Y)^{-1}$. Any assistance would be greatly appreciably.

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You can prove the result directly by playing with the difference quotient a bit. Let's write $$ (X+H)^{-1} - X^{-1} + X^{-1}H X^{-1} =X^{-1} [ X(X+H)^{-1}- I + H X^{-1} ] \\ =X^{-1}[ (X - (X+H)) (X + H)^{-1} + H X^{-1}] \\ = X^{-1}[H (X+H)^{-1} + H X^{-1}] = X^{-1}H [(X+H)^{-1} - X^{-1}]. $$ With this in hand you can get the result directly from the continuity of the inverse map.

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Note that the Frechet derivative of a function at a point is unique. By definition:

$$(X+Y)^{-1} - X^{-1} = D\Phi(X)(Y) + o(\|Y\|^2)$$

We can handle the calculation of $(X+Y)^{-1}$ by writing its Taylor series:

$$\frac{1}{X+Y}=\frac{1}{(I+YX^{-1})X}=X^{-1}(I-YX^{-1}+o(\|Y\|^2))$$

The last step is true because $(AB)^{-1}=B^{-1}A^{-1}$. Also, the inversion Taylor series is valid when $\|Y\|<1$. But since $Y \to 0$, there's no problem. Therefore,

$$(X+Y)^{-1}-X^{-1}=-X^{-1}YX^{-1}+o(\|Y\|^2)$$

This shows that $-X^{-1}YX^{-1}$ indeed fits in the definition of the Frechet derivative. Hence, $D\Phi(X)(Y)=-X^{-1}YX^{-1}$ because of uniqueness. $\fbox{Q.E.D.}$

ADDENDUM: More generally, one can show, using the same approach, that when $\Phi(X)=X^{-\alpha}$ where $\alpha \in\mathbb{N}$, we have $$D\Phi(X)(Y)=-\alpha X^{-\alpha}YX^{-1}$$ A beautiful generalization of the commutative case that we have seen before in the calculus of real and complex numbers. With appropriate care, one may handle the cases $\alpha \in \mathbb{R}$ or $\alpha \in \mathbb{C}$ for when $X$ is an element any Banach algebra.

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