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I was wondering how one computes the derivative of the inverse of a matrix wrt to the original matrix. In my case, it might even be simpler as the matrix is an invertible diagonal matrix.

So, $X$ is a diagonal matrix and I would like to compute the derivative $\frac{dX^{-1}}{dX}$

I was wondering if the result should be the diagonal matrix, but could not convince myself why that should be.

Luca
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  • Do you mean that you want the derivative of the function $inv: GL_n(\mathbb{R})\rightarrow GL_n(\mathbb{R})$ such that $inv(A):=A^{-1}$? Try to define a topology (easier- a metric) on $GL_n(\mathbb{R})$ and apply the definition of derivative... – marco trevi Sep 25 '14 at 10:08
  • See also this: http://math.stackexchange.com/questions/128556/inversion-of-matrices-is-a-diffeomorphism – marco trevi Sep 25 '14 at 10:13

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I'm not sure this helps, but note that

$0=d(X X^{-1})=(dX)X^{-1}+Xd(X^{-1})\quad\implies\quad d(X^{-1})=-X^{-1}\cdot dX\cdot X^{-1}$.

If $X$ is diagonal, I guess you can write it as

$\dfrac{dX^{-1}}{dX}=-X^{-2}$

For example, if $X$ is $1\times 1$, a.k.a. a number, this is just

$\dfrac{d(x^{-1})}{d x}=-x^{-2}$

davyjones
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  • Thanks for that. Yes, perhaps, this is what I was looking for! Does this mean I can simply multiply the inverse of the matrix with itself. – Luca Sep 25 '14 at 10:27
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    @Luca, it is fine as long as the matrix is diagonal. You can treat them separately as the usual $1\times 1$ case. For general matrices, you cannot do that. – davyjones Sep 25 '14 at 11:28