The Jordan and Frobenius pairs of the inverse

Given a matrix A ∈ Cn×n there exists a nonsingular matrix V such that V−1AV = 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 Vˆ and ˆJ such that Vˆ −1A−1Vˆ = ˆJ and ˆJ 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.