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Given a matrixthere exists a nonsingular matrixsuch that, where  J  is a very sparse matrix with a diagonal block structure, known as the Jordan canonical form (JCF) of. Assume thatis nonsingular and thatandare given. How to obtainandsuch thatandis the JCF of? 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.

The Jordan and Frobenius pairs of the inverse