Leverrier’s Algorithm: A New Proof and Extensions
Stephen Mark Barnett · SIAM Journal on Matrix Analysis and Applications · 1989
A new derivation is given of the Leverrier–Fadeev algorithm for simultaneous determination of the adjoint and determinant of the $n \times n$ characteristic matrix $\lambda I_n - A$. The proof uses an appropriate companion matrix and is of some interest in its own right. The method is extended to produce a corresponding scheme for the inverse of the polynomial matrix $\lambda ^2 I_n - \lambda A_1 - A_2 $, and indeed can be generalized for a regular polynomial matrix of arbitrary degree. The results.have application to linear control systems theory.