Is there an analytic form to find the invers of a matrix using jordan form?

Question

I know that any cyclic matrix can be inversed by diagonalaizing it $A = PDP^{-1}$ where the columns of $P$ are Fourier basis vectors and the diagonal of the (diagonal matrix) $D$ contains the eigenvalues of $A$ which are the discrete Fourier transrom on the elements of the generating vector of $A$. In this case we have a nice analytic formula for the inverse of $A$:

$$A^{-1} = PD^{-1}P^{-1}$$

My question is:

Given an invertible matrix, is there an analytic formula for it's inverse using it's Jordan form? I am looking for something like the above equation which might be applied on not-necessarily cyclic matrices.

Edit: Also, Is there a canonical form for $J^{-1}$?

Thank you!

Answer 1

$$A=C^{-1}JC \Rightarrow A^{-1}=C^{-1}J^{-1}C$$