Advances in Linear Algebra & Matrix Theory
Vol.2 No.4(2012), Article ID:25472,4 pages DOI:10.4236/alamt.2012.24008
On the Minimal Polynomial of a Vector
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
Keywords: Finite dimensional linear space; Linear transformation; Minimal polynomial
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 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.
1. Introduction
Let be a field,
be a vector space over
with dimension
, and
be a linear transformation of
. It is known that
becomes a
-module according to the following definition:
.
For a fixed linear transformation and a vector
, the annihilator of
with respective to
is defined to be
.
Similarly, the annihilator of with respective to
is defined to be
.
Since 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
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:
, (1)
and are vectors in
such that
(2)
where. Let
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
.
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 arbitrary 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 an infinite field can not be an union of a finite number of its proper subspaces by Vandermonde determinants.
Lemma 1. Let be an infinite field, and
be a vector space over
with dimension
, and
be nontrivial subspaces of
for
. Then there exists infinite many bases of
such that any element of them is not in each
for
. Therefore, if
then
for some i .
Proof: Let be a F-base of
. For any
we set
.
Let distinct elements in
. We have
where is a Vandemonde matrix. So
is a base of
because the determinant of
is nonzero. Let
be the following set with an infinite number of vectors:
.
Since with
is a nontrivial subspace of
one can verify that
. And so
.
Therefore, is infinite, and any distinct
vectors in the set constitute a base of
.
Proposition 1. Let be an infinite field. Let
be a
-vector space with dimension
, and
be a linear transformation of
. Then there exists a vector
such that
.
Proof: It is clear that are linearly dependent over
. So the degree
. For any
, the minimal polynomial
of
is a monic factor of
. So there exist finite number of vectors
such that
where
are mutually coprime irreducible polynomials. Set
. One can verify that
.
By Lemma 1, there exists with
such that
. Which shows that
and so
is a zero linear transformation. Hence we have
.
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 be a field,
be a
-dimensional linear space over
, and
be a linear transformation of
. For any
, there exists
such that
here
and the following
stand for the least common multiple and greatest common divisor of two polynomials, respectively.
Proof: By properly arrangement, the minimal polynomials of with respective to
have the following irreducible factorization respectively,
,
.
Moreover, for
, and
for
. So, we have
,
.
One can verify that the minimal polynomials of and
are
respectively. Set
, then
.
Which implies that
. (3)
Conversely, from it follows that
.
Which shows that
So, since
Similarly,
. By
again, we have
. (4)
Equations (3) and (4) imply that
.
Proposition 2. Let be a field. Let
be a
-vector space with dimension
and
be a linear transform of
. Then there exists a vector
such that
.
Proof: Let be a
-base of
. One can verify that
.
By repeatedly utilization of Lemma 2, we can find a vector such that
According to Proposition 2, we can easily deduce the Cayley-Hamilton theorem.
Proof of Cayley-Hamilton Theorem: Let be the characteristic polynomial of
. We show
. By Proposition 2 there exists
such that
. Let
.
So, one can verify that vectors are linearly independent over
. We extend them to a basis of
as follows:
.
We have
where the square matrix
has the form
and is an
square matrix, and
is an
matrix. So the characteristic polynomial of
is
and
.
Hence, , and
.
Actually, the Cayley-Hamilton theorem can be obtained by only using the minimal polynomial of a vector.
Another Proof of Cayley-Hamilton Theorem: Let be the characteristic polynomial of
. For any
let
be the minimal polynomial of the vector
with respective to
. To prove the CayleyHamilton theorem, it is enough to show that
.
This statement can be verified by the same arguments as that in above proof.
3. Acknowledgements
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.
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