Basic Modern Algebra with Applications by Mahima Ranjan Adhikari & Avishek Adhikari

Author:Mahima Ranjan Adhikari & Avishek Adhikari
Language: eng
Format: epub
Publisher: Springer India, New Delhi

The corollary leads to the following definition.

Definition 8.5.12

Let V be an n-dimensional vector space over F and T be a non-zero element in L(V,V). Then there exists a non-trivial polynomial m(x) of lowest degree with leading coefficient 1 in F[x] such that m(T)=0. We call m(x) minimal polynomial for T over F.

Remark

If T satisfies a minimal polynomial g(x) in F[x], then this m(x) divides g(x) in F[x]. Since g(x) is monic and g(T)=0, m(x)=g(x). This shows the uniqueness of a monic polynomial for T over F and we use the term the minimal polynomial for T over F.

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