Theory of a linear transformation

Abstract

Consider the algebra A(E; E) of linear transformations and fix an element φ∈A(E; E). Then a homomorphism Φ: Γ [t]→A(E; E) is defined by

$$\Phi :f \mapsto f(\varphi )$$

(cf. sec. 12.11). Let μ be the minimum polynomial of φ. Since A(E; E) is non-trivial and has finite dimension, it follows that deg μ ≧ 1 (cf. sec. 12.11). The minimum polynomial of the zero transformation is t whereas the minimum polynomial of the identity map is t-1.

In this chapter E will denote a finite-dimensional non-trivial vector space defined over an arbitrary field Γ of characteristic 0, and φ: E→E will denote a linear transformation.