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.

  • $\begingroup$ 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


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 $(H,K)\mapsto HK+KH$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.