Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am trying to find the derivative of a matrix with respect to the inverse of the same matrix. The matrix in question is a non singular symmetric matrix. Any thoughts?

share|improve this question

1 Answer 1

The starting point is the von Neumann series $$(I-A)^{-1}=I+A+A^2+\dots$$ which implies that the matrix function $A\mapsto (I-A)$ has identity as its derivative at $A=0$.

The general case of $B\mapsto B^{-1}$ differentiated at $B_0$ can be reduced to the above by considering $B=(I-A)B_0$, hence $B^{-1}=B_0^{-1}(I-A)^{-1}$. The linear term here is $B_0^{-1}A$, and since $A=-(B-B_0)B_0^{-1}$, the derivative can be written as $\Delta B\mapsto -B_0^{-1}(\Delta B) B_0^{-1}$.

Which, not incidentally, simplifies to $(1/x)'=-1/x^2$ when everything commutes.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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