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

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