Gapplet

Let A be a square matrix over Q of dimension n. It generates a subalgebra, say S of the ring of all square matrices of dimension n. If d Q[X] is its minimum polynomial, then there is a morphism f: S -> Q[X]/(d) given by

where c_i Q and x = X + (d) Q[X]/(d).

Thus, we find a morphism f : S -> Q[X]/(d). if we give the minimum polynomial of a matrix, and that is what we do here.

The morphism is actually an isomorphism with inverse f^-1 : g -> g(A). It can be used to find an inverse of A, as follows. Compute the inverse of f(A) in Q[X]/(d). This will be represented by a polynomial g Q[X] of degree less than n. Now g(A) is the inverse of A