The inverse of $A - A^{-1}$

Question

I've been trying to calculate the inverse of $A - A^{-1}$ but I couldn't. I'm wondering whether there's a closed form solution for the inverse of $A - A^{-1}$ . If not, is there any good approximation for it?

Answer 1

As mentioned in the comment, $A-A^{-1}$ may not be invertible. However we still have

$$A(A-A^{-1}) = A^2 - I = (A-I)(A+I)$$

therefore if $(A-I),(A+I)$ are invertible so is $(A-A^{-1})$ and the inverse is $$(A-I)^{-1}(A+I)^{-1}A$$

Answer 2

Note that $$(A-A^{-1})(-A-A^3-A^5-...) =I$$ Thus if $$B=\sum _{n=1}^\infty (-A)^{2n-1}$$ converges, it is the inverse of $A-A^{-1}$