An algorithm for computing Aⁿ

Abstract

The space of all m × m matrices is an m²-dimensional linear space. In this section A is any m × m matrix,real or complex. This means there is a smallest integer r such that I,A,…,A^r-1 are linearly independent, and hence there are real numbers α_r-l,α_r-2,…,α₀ such that

$${A^r} + {\alpha _{r - 1}}{A^{r - 1}} + \cdots + {\alpha _0}I = 0$$

.