On the Minimal Polynomial of a Vector

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 v 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.


Introduction
Let F be a field, V be a vector space over F with dimension , and n  be a linear transformation of .It is known that becomes a V V   F x -module according to the following definition: For a fixed linear transformation  and a vector , the annihilator of with respective to Similarly, the annihilator of with respective to V  is defined to be is a principal ideal domain the ideals and can be generated by the unique monic polynomials, denote them by and , respectively.Which are called the order ideals of and in abstract algebras, respectively.They are also called the minimal polynomials of and V with respective to  in linear algebras, respectively.It is clear that the minimal polynomial of zero vector (or zero transformation) is 1.By the structure theorem for finitely generated modules over a principal ideal domain [3,4], the module can be decomposed into a direct sum of finite cyclic submodules: where 1, 2, , 1 be the characteristic polynomial of  .By (1) and (2) one has Furthermore, these results straightforwardly imply the following theorem: Theorem 1. [3,4] With the notations as above, we have 1) [Cayley-Hamilton Theorem]  , and so .
2) There exists a vector such that an proposition in .

Proofs Based on Linear Algebras
In this section we give an alternative proof of Theorem 1 by only utilization of knowledge of linear algebras.To demonstrate an interesting proof of some proposition in linear algebras and its applications, we present two proofs of (2) in Theorem 1 for infinite fields and arbitrary fields, respectively, and then use the related results to prove the Cayley-Hamilton theorem.
The following lemma provide an interesting proof of linear algebras that a vector space over an infinite field can not be an union of a finite number of its proper subspaces by Vandermonde determinants.Lemma 1.Let F be an infinite field, and V be a vector space over F with dimension n , and i V be nontrivial subspaces f V for 1, 2, i s   .Then ere exists infinite many bases of V any element of them is not in each Therefore,  is infinite, and any distinct vectors in the set constitute a base of Let F be an infinite field.Let be a V F -vector space with imension n , and d  be linear transformation of V .Then t re exists a vector , where are mutually coprime irreducible poly-For any v V  , the minimal polynom is a monic factor of By Lemma 1, there exists with such th and so Moreover, for , and One can verify that the minima polynomials of l Conversely, from Which shows that , . .
Equations ( 3) and ( 4) imply that F be a field.Let be a V Copyright © 2012 SciRes.ALAMT F -vector space with mension n and and is an C n m  square matrix, and X is an   Hence, Actually, the Cayley-Hamilton theorem can be obtained by only using the minimal polynomial of a vector., and .

 
This statement can be verified by the same arguments as that in above proof.
polynomial of the vector with respective to v  .To prove the Cayley-Hamilton theorem, it is enough to show that yley-Hamilton theorem.Proof of Cayley-Hamilton   x   be the characteristic polynomial of  .We show . Acknowledgements are linearly independent over F .W basis of V as follows: like to thank the anonymous referees for helpful comments.The work of both authors was supported by the Fund of Linear Algebras Quality Course of Hubei Province of China.The work of D. Zheng was supported by the National Natural since Foundation of China (NSFC) under Grant 11101131.