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Let $U = \{X \in M_n \mid I-X \textrm{ is invertible}\}$. Show $f(X) = (I-X)^{-1}$ is twice differentiable at $0$, and compute its first and second derivatives there.

$M_n$ refers to real $n \times n$ matrices. For context, earlier parts of the question asked me to show $U$ is open in $M_n$ and show $U$ contains $V = \{X \in M_n \mid ||X|| < 1\}$. I was also asked to show that the series $\sum_{k=0}^\infty X^k$ converges to $f$ on $V$. I have finished all these parts, and only the question above remains.

Progress so far: I have found the first derivative at $0$, which was $I$. To find the second derivative I expect I'm to compute the first derivative in a neighbourhood of $0$ first, and differentiate that, but I can't get this to work, neither using the explicit expression for $f$ or the series.

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  • $\begingroup$ This identity is usually referred to as the Neuman series. I don't quite see which of the additional properties you have solved already. It's worth noting that $0 \in V$ and maybe the series representation of $f$ is easier to derive? $\endgroup$
    – Roland
    Jun 3, 2017 at 18:34
  • $\begingroup$ I've solved all the additional properties I've written, the only thing left is to show the second derivative exists and find it. $\endgroup$
    – B. Mehta
    Jun 3, 2017 at 18:37

1 Answer 1

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When $\|X\|<1$, expand $f(X)=(I-X)^{-1}$ to the power series $I+X+X^2+\cdots$. The $O(H)$ term of $f(X+H)-f(X)$ is thus $Df(X)(H)=H+(HX+XH)+(HXX+XHX+XXH)+\cdots$ and the $O(K)$ term of $Df(X+K)(H)-Df(X)(H)$ is $$ (HK+KH)+(HKX+HXK+KHX+XHK+KXH+XKH)+\cdots. $$ Hence the derivative of $f$ at $X=0$ is $H\mapsto H$ and the second derivative at zero is $(H,K)\mapsto HK+KH$.

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