商空间诱导变换的最小多项式

Minimal Polynomial of Induced Transformation on Quotient Space

摘要: 本文给出了商空间诱导变换的定义，推导了诱导变换的基本性质；针对诱导变换的最小多项式，证明其整除原线性变换最小多项式的核心定理，并阐明其与子空间诱导变换最小多项式的最小公倍式关系；分析了诱导变换最小多项式与若尔当标准形的内在联系；最后给出典型应用。

Abstract: This paper presents the definition of the induced transformation of a quotient space and deduces the basic properties of induced transformations. For the minimal polynomial of the induced transformation, it proves the core theorem that it divides the minimal polynomial of the original linear transformation, clarifies its relationship with the least common multiple of the minimal polynomials of induced transformations on subspaces, analyzes the intrinsic connection between the minimal polynomial of induced transformations and the Jordan canonical form; and finally provides typical applications.