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I am having problems showing that the function $ \operatorname{inv}:G\rightarrow G$ $A\rightarrow A^{-1}$ where $G$ is the set of all invertible $n\times n$ matrices, is a diffeomorphism. I have already shown that such function is a homeomorphism, and its inverse is itself, but I don't know how I can show that this function is differentiable.

The exercise also tells us that the derivative of $\operatorname{inv}$ in $A$ is the linear mapping $M\rightarrow M$ such that $X\rightarrow -A^{-1}\cdot X\cdot A^{-1}$.

Can anybody give me a hint?

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    How did you prove the $\text{inv}$ is a homeomorphism ?2015-03-08

2 Answers 2

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Here is a hint: what does Cramer's rule tell you about the matrix entries of the inverse in terms of the original matrix?

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    Thanks @Michael, I didn't know that det was a LaTeX function!2012-04-06
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Another approach would be to note that if $\|X\|<1$, then $(I+X)^{-1} = I-X+X^2-\cdots$. Then consider the expression $(A+X)^{-1}$, where $X$ is a suitably small perturbation. A small computation shows that $(A+X)^{-1} = (I+A^{-1}X)^{-1} A^{-1}$. Then expand using the above series, and look at the linear term of the expansion. Both differentiability and the form of the derivative follow from this expansion.

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    If you take $X \in \mathbb{R}$, you can see that the series fails to converge if $|X| \geq 1$.2012-04-06