TITLE:
On the Minimal Polynomial of a Vector
AUTHORS:
Dabin Zheng, Heguo Liu
KEYWORDS:
Finite Dimensional Linear Space; Linear Transformation; Minimal Polynomial
JOURNAL NAME:
Advances in Linear Algebra & Matrix Theory,
Vol.2 No.4,
December
17,
2012
ABSTRACT: It is well known that the Cayley-Hamilton theorem is an interesting and important theorem in linear algebras, which was first explicitly stated by A. Cayley and W. R. Hamilton about in 1858, but the first general proof was published in 1878 by G. Frobenius, and numerous others have appeared since then, for example see [1,2]. From the structure theorem for finitely generated modules over a principal ideal domain it straightforwardly follows the Cayley-Hamilton theorem and the proposition that there exists a vector v in a finite dimensional linear space V such that and a linear transformation of V have the same minimal polynomial. In this note, we provide alternative proofs of these results by only utilizing the knowledge of linear algebras.