On the Structural Relationship Between the Characteristic and Minimal Polynomials of a Linear Operator

Abstract

In this paper, we study the relationship between the characteristic and minimal polynomial of a linear operator, with a focus on figuring out under what conditions that the two polynomials equal each other.

We emphasize that the characteristic and minimal polynomial of a linear operator are the same if and only if every eigenvalue has a geometric multiplicity of 1, which is equivalent to having only one Jordan block per eigenvalue. We provide an alternative proof for such a theory. For such matrices, we also show that the minimal polynomial can be easily derived from the normalized linear dependence of the Krylov sequence $\{v, Av, A^2v, \dots, A^{n-1}v\}$ for any generic vector $v$.

We apply these algorithms to analyze the nilpotent and companion matrices. The results algorithmically verify that for a companion matrix $C$, its characteristic and minimal polynomials are identical and equal to its generating polynomial, $p_C(X)=m_C(X)=f(X)$. For a nilpotent matrix $N$ with index $k$, we confirm that its minimal polynomial is $m_N(X)=X^k$.