Another algorithm for computing Aⁿ.

Abstract

In Section 4 we gave an algorithm for computing Aⁿ that depended upon computing the eigenvalues of A. Here in this section we give an algorithm that does not require computing the eigenvalues. As before we let

$$\psi \left( \lambda \right) = {\lambda ^s} + {a_{s - 1}}{\lambda ^{s - 1}} + \cdots + {a_0}$$

be any polynomial that annihilates A -- i.e., such that ψ(A) = 0. We can, for instance, always take ψ(λ) to be the characteristic polynomial of A.