Minimum and Characteristic Polynomials of Low-Rank Matrices
William P. Wardlaw · Mathematics Magazine · 1995
Introduction Given an n X n matrix A, it can be useful to have a monic polynomial p(x) of low degree that annihilates A; that is, for which p(A) = 0. For example, higher degree polynomials in A can be evaluated by reducing them modulo p(x). A low degree annihilator of A can be used in algorithms to calculate the matrix e At, as in [1, Thm. 6.10, p. 284]. Moreover, knowledge of such an annihilator can simplify the determination of the minimum polynomial or the null ideal of A. A simple dimension argument shows that the set {Ak: 0 < k < n2} is dependent, and hence that A satisfies a polynomial equation