On the Minimal Polynomial of a Vector

Advances in Linear Algebra & Matrix Theory, 2012, 2, 48-50

doi:10.4236/alamt.2012.24008 Published Online December 2012 (http://www.SciRP.org/journal/alamt)

On the Minimal Polynomial of a Vector

Dabin Zheng, Heguo Liu

Faculty of Mathematics and Computer Science, Hubei University, Wuhan, China Email: dzheng@hubu.edu.cn, ghliu@hubu.edu.cn

Received October 26, 2012; revised November 30, 2012; accepted December 9, 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 theo- rem 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.

Keywords: Finite Dimensional Linear Space; Linear Transformation; Minimal Polynomial

1. Introduction

Let

F

be a field,

V

be a vector space over

F

with dimension n , and  be a linear transformation of . It is known that becomes a

V V ^{F x}

  -module according to the following definition:

 

F x  V V

 ^{f x}   ^{, v} 

^

^f   ^{ v .}

For a fixed linear transformation  and a vector , the annihilator of with respective to

vV

v  is de-

fined to be

        

⁰

ann v  p x F x p



v

 ^.

Similarly, the annihilator of

V

with respective to  is defined to be

        

^{0 ,}

ann V  p x F x p



v  v V

 ^.

Since

^{F x}

  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

V

in abstract algebras, respectively. They are also called the minimal polynomials of and

V

with respective to

 

ann v

 

m_ x

v

 

ann V

 

mv x

 in linear algebras, respectively. It is v 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:

V

 

1

 

2

 

s

V F x



F x



  F x

 , (1) and  

₁, ₂,,



_s

are vectors in

V

such that

 

_{i i}

  ,

_i

 

_i1

 

ann   d x d x d

_

x , s

(2) where i  1, 2,



,  1 . Let 

_

  x be the characteristic polynomial of  . By (1) and (2) one has



m

_

  x  ann   

s

 d

s

  x ;



     

1 1

s s

i i

i i

x ann d x





 

 





 ^.

Furthermore, these results straightforwardly imply the following theorem:

Theorem 1 [3,4] With the notations as above, we have 1) [Cayley-Hamilton Theorem]

   

m

_

x 

_

x , and so 



   

⁰

. 2) There exists a vector v  V such that

  

mv x m_ x

 .

2. 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 arbi- trary fields, respectively, and then use the related results to prove the Cayley-Hamilton theorem.

The following lemma provide an interesting proof of an proposition in linear algebras that a vector space over

Copyright © 2012 SciRes. ALAMT

D. B. ZHENG, H. G. LIU 49

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 be a vector space over

V F with dimension , and V

_i

be nontrivial subspaces of for i . Then there exists infinite many bases of such that any element of them is not in each for . Therefore, if

then

V

for some i . n



, , V

V

i

Vi

, _n

1, 2,

 1, 2, i 



s V

s

1V

i

Proof: Let

s

V 



i ^

1, 2,

  

n



be a F-base of . For any we set

V

b  F

2 1

1 2 3

n

b b b b















  ^

 . Let

b b₁, ₂,,b_n

distinct elements in

F

. We have



1, 2, ,

 ^

1, 2, ,

^{ }

1, 2, ,

b b bn n Van b b bn

 







 





^





where is a Vandemonde matrix. So

1 2

b b



1, 2, , _n Van b b b

, , ,

bn

 







b b1, 2,,b_n

is a base of because the determinant of is nonzero. Let be the follow- ing set with an infinite number of vectors:



V

Van S



b



S 



bF s

.

Since

_i

with is a nontrivial subspace of one can verify that

V i1, 2,,

V SV_i  n 1

. And so

   

1 1

1

s s

i i

i i

S V S V s n

 

   

 

 



 

 ^

 

.

Therefore, is infinite, and any distinct

1

\

s i i

S S V



 

 





n vectors in the set constitute a base of

V

.

Proposition 1. Let

F

be an infinite field. Let be a

V F

-vector space with dimension n , and  be a lin- ear transformation of

V

. Then there exists a vector

such that

vV v

 

m_

 

x , 2, m x 

.

Proof: It is clear that

1,

 

,



ⁿ²

are linearly de- pendent over

F

. So the degree ^deg 

^m

 

^x



ⁿ²

. For any , the minimal polynomial of is a monic factor of . So there exist finite number of

vectors such that

vV v ii

 

^x

^v

mv

 

m_ x

2,



,

s

,



1,

 

v₁

 

v₂

 

v_s

 

m_ x m x m x m x

, where

vi

are mutually coprime irreducible poly- nomials. Set

 

m x s 



ⁱ

  ⁰ 

i v

V    V m    . One can verify that

1 2 s

V  V



V



V .

By Lemma 1, there exists with such that . Which shows that

k 1   k s V  V

k

  0, for all

vk

m      V ,

and so m

v_k

   is a zero linear transformation. Hence we have ^m   ^x  ^m



  ^x .

vk

In fact, Proposition 1 holds for arbitrary fields from

the introduction. To obtain a general proof we first give the following lemma.

Lemma 2. Let

F

be a field, be a -dimensional linear space over

V n

F

, and  be a linear transformation of V . For any

0

 

, V

, there exists   V such that

     ^,   

m

_

x  lcm m

_

x m

_

x ,

here and the following stand for the least common multiple and greatest common divisor of two polynomials, respectively.

lcm gcd

Proof: By properly arrangement, the minimal poly- nomials of  

,

with respective to  have the fol- lowing irreducible factorization respectively,

 

   

1 1

1 2

1 ^{r r}1 ^s

k

k k k

r r s

u x u x

m

_

x 



p



p

 

p

_^ 

p ,

 

   

1 1

1 2

1 1

s

r r l

l l l

r r s

v x v x

m

_

x  p



p p

_^ 

p

 

.

Moreover, k

_i

 l

_i

for i  1, 2,



, r , and k

_i

 l

_i

for 1, ,

i   r



s . So, we have

   

 ^, 

¹

^{   }

²

lcm m

_

x m

_

x  u x v x ,

   



1 2



gcd u x v , x  1 .

One can verify that the minimal polynomials of

2

 

u   and v

1

    are

 

   

_{ }

   

2 1

,

1 2

u v

m

_{ }

x  u x m

_{ }

x  v x , respectively. Set ^{ } u

2

  ^{  } v

1

 ^{ }  ^{, then}

   

1 2 0

u



v

 



. Which implies that

     

1 2

m

_

x u x v x . (3) Conversely, from u

2

  ^{    } v

1

 ^{ }  it follows that

 

_{v1 }

    

2



0 m_



m _{ }



u

 



. Which shows that

 

   

_{ }

       

2 1 , . . 1 2

u v

m _{ } x m_ x m _{ } x i e u x m_ x v x

So,

u x m1

 

_

 

x

since

gcd



u x v1

   

, 2 x



1

Simi- larly,

v2

 

x m_

 

x

. By

gcd



u x v1

   

, 2 x



^1

^again, we have

     

1 2

u x v x m

_

x . (4) Equations (3) and (4) imply that

 

¹

   

²

     

m_ x u x v x lcm m_ x m_ x

. Proposition 2. Let

F

be a field. Let

V

be a

Copyright © 2012 SciRes. ALAMT

D. B. ZHENG, H. G. LIU

Copyright © 2012 SciRes. ALAMT 50

F

-vector space with dimension n and  be a linear transform of

V

. Then there exists a vector

vV

such that

and

C

is an n m  square matrix, and

X

is an



^m



m n

matrix. So the characteristic polynomial of

 is

     

n 0 m

B X mv x m_ x

.

x xI xI B xIn m C

 C 

 

      

 

,

Proof: Let  

₁, ₂,,



_n

be a

F

-base of

V

. One

can verify that and

 

₁

 

₂

m_ x lcm m_ x m^, _

 

^{x ,}

 





 

^x





mv

,mn

 

^x





V lcm

1

1 1

m m

m m 0

xI B x b _x ^   b x b

. .

Hence,

^m

   

^x  ^x

, and



  

⁰

. By repeatedly utilization of Lemma 2, we can find a

vector

v

such that

  

1

 

, 2

 

, ,

^

m_ x  m_ x m x mn



^ x ^.

Actually, the Cayley-Hamilton theorem can be ob- tained by only using the minimal polynomial of a vector.

Another Proof of Cayley-Hamilton Theorem: Let

 

^x



be the characteristic polynomial of  . For any

vV

let

mv

 

x

be the minimal polynomial of the vector v with respective to  . To prove the Cayley- Hamilton theorem, it is enough to show that

According to Proposition 2, we can easily deduce the Cayley-Hamilton theorem.

 

Proof of Cayley-Hamilton Theorem: Let

_ x

be the characteristic polynomial of  . We show

    m

_v

    x 

_

x for any v  V .

m

_

x 

_

x . By Proposition 2 there exists

vV

such that

^m

 

^x

. Let

m

v m_

 

x

 

v

 

m

m_ x  x x^mb _1x^m1 b₁

 

x b0

 

This statement can be verified by the same arguments as that in above proof.

. So, one can verify that vectors

^v,

^

^{v ,,}

^

^m1

 

^v

are linearly independent over

F

. We extend them to a basis of

V

as follows:

3. Acknowledgements

 

¹

 

, , , ^m , ,

v



v 



^ v ,



₁

, ,

n m

n m





0 0 0 1

 







n m_

B X C

. We have

  

   

1 1

1 1

, , , ,

0

m

m

v v

v v

   

  



 

 

 



 

 

 





 , ,

, ,

The authors would 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.

REFERENCES

where the m square matrix

B

has the form

0 1 2

1

0 0

1 0

0 1

0 0 _m

b b

B b

b _

 

  

 

 



 







   



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[4] Th. W. Hungerford, “Algebra, GTM 73,” Springer-Verlag, New York, 1980. doi:10.1007/978-1-4612-6101-8