The Jordan and Frobenius pairs of the inverse

Given a matrix A is an element of C-nxn there exists a nonsingular matrix V such that V-1 AV = J, where J is a very sparse matrix with a diagonal block structure, known as the Jordan canonical form (JCF) of A. Assume that A is nonsingular and that V and J are given. How to obtain (V) over cap and (J) over cap such that (V) over cap (-1) A(-1) (V) over cap = (J) over cap and (J) over cap is the JCF of A(-1) ? Curiously, the answer involves the Pascal matrix. For the Frobenius canonical form (FCF), where blocks are companion matrices, the analogous question has a very simple answer. Jordan blocks and companions are non-derogatory lower Hessenberg matrices. The answers to the two questions will be obtained by solving two linear matrix equations involving these matrices