Jordan Canonical Form with Parameters from Frobenius Form with Parameters

The Jordan canonical form (JCF) of a square matrix is a foundational tool in matrix analysis. If the matrix A is known exactly, symbolic computation of the JCF is possible though expensive. When the matrix contains parameters, exact computation requires either a potentially very expensive case discussion, significant expression swell or both. For this reason, no current computer algebra system (CAS) of which we are aware will compute a case discussion for the JCF of a matrix A(α) where α is a (vector of) parameter(s). This problem is extremely difficult in general, even though the JCF is encountered early in most curricula. In this paper we make some progress towards a practical solution. We base our computation of the JCF of A(α) on the theory of regular chains and present an implementation built on the RegularChains library of the Maple CAS. Our algorithm takes as input a matrix in Frobenius (rational) canonical form where the entries are (multivariate) polynomials in the parameter(s). We do not solve the problem in full generality, but our approach is useful for solving some examples of interest.