1. Introduction
Linear algebra has a wide range of application in most scientific and technological areas. As a fundamental part, the theory of vector space and linear map has developed to be a nearly perfect subject since nineteenth century. In the viewpoint of the author, when applying the theory to diverse problems, the more the detailed aspects of the knowledge one is in possession of, the more proficient one will become in dealing with those related problems. For example, in one of the excellent dynamical system monographs “Differential equations, Dynamical Systems, and an Introduction to Chaos” [1] , the authors put a separate chapter named “Higher Dimensional Linear Algebra” to help the readers recall some relevant linear algebra knowledge needed for the analysis of differential equations and dynamical systems, despite so many linear algebra monographs available for reference, such as [2] -[8] to name a few. Although it is just a brief review of higher dimensional linear algebra, the authors presented therein a distinct method to prove that a 
matrix can be transformed by a similarity transformation to a Jordan block matrix when it has only one eigenvalue and one-dimensional eigenspace. The essence of the method resides in a subtle analysis of the dimension of the kernel and the image of the matrix. Motivated by this method, this study is intended to generalize the results to the 
matrix. For this purpose, it is found that it is necessary to gain a deep insight into some geometric properties of the 
subspace, which finally leads to a more specific relationship discovered between the power-m kernel and the power-m image of linear map
, which is reffered to as the Cross-Identity theorem below. As a corollary of this theorem, we proved the fact that a 
square matrix with only one eigenvalue and one eigenvector is similar to a Jordan block matrix.
The paper is organized as follows. In Section 2, some basic facts will be recalled as preliminaries for the good understanding of the work. Section 3 presents the main results of this paper with some lemmas preceding the Cross-Identity theorem. Some conclusions are drawn finally in Section 4.
2. Preliminaries
Let 
be a n-dimensional vector space over the complex numbers, and 
be the set of all linear maps mapping the vector space 
into itself, and let 
and 
denote the kernel (null space) and the image (range) of a linear map
, respectively, then it is well known that if the characteristic polynomial of 
is factorized as
, 
being its distinct eigenvalues, then the vector space 
can be decomposed into the direct sum of some subspaces, i.e.
, where 
with 
denoting the identity map. Moreover, it can also be proved by using the primary decomposition that each of the above subspace
, and hence the whole vector space
, can be further expressed as a direct sum of 
cyclic subspaces [3] . Because the transformation matrix of a linear map under a cyclic basis takes on a Jordan normal form, the above decomposition theories lead straightforward to a geometric approach of proof for the fact that any square matrix 
can be brought into Jordan normal form through a similarity transformation with a change-of-basis matrix (another way leading to this result is through the elementary divisor theory [7] ).
From the above viewpoint, the cyclic subspace of a vector space shall be an important concept in the field of linear algebra. For the convenience of reference, some preliminary concepts are presented below, for which one can also refer to [3] .
Definition 2.1 Cyclic vector Let 
be a complex number, let
, 
, then 
is called 
if there exists an integer 
such that
. The smallest positive integer 
having this property will then be called a period of 
relative to
.
Definition 2.2 Cyclic space The vector space 
will be called cyclic if there exists some number 
and an element which is 
and 
generate
.
Theorem 2.3 if 
is
, with period
, then 
are linearly independent.
Definition 2.4 Cyclic basis if 
is
, with period
, then 
is called a cyclic basis or Jodan basis for the cyclic space 
it generates.
Definition 2.5 Jordan block matrix A matrix is called a Jordan block matrix if it has the same number 
on the diagonal, 1 above the diagonal, and 0 everywhere else.
Theorem 2.6 Let 
be a linear map defined on a 
-dimensional cyclic space
, then the associated matrix 
with respect to the cyclic basis 
is equal to a Jordan block matrix.
Hence, it has only one eigenvalue 
and one-dimensional eigenspace.
Definition 2.7 Fan and Fan basis By a fan of linear map A in vector space 
, we shall mean a sequence of subspaces 
such that 
is contained in 
for each
, such that
, and finally such that each 
is
. By a fan basis, we shall mean a basis 
of 
such that 
is a basis for
.
3. Cross-Identity Theorem
To state the results of this paper, some more notations need to be introduced. As is known, the customary eigenspace of a linear map 
associated with eigenvalue 
is equivalent to
, and the corresponding generalized eigenspaces are the union set of
. For convenience, we refer to 
and 
as power-m kernel and power-m image with respect to linear map 
, respectively. In this case, the customary eigenspace and the generalized eigenspaces associated with eigenvalue 
are the power-one kernel and the higher power kernels of linear map
.
Since a vector space can be expressed as a direct sum of 
cyclic subspaces, we only consider one of its 
subspace, or without loss of generality, suppose 
entirely is a 
space hereafter. In this situation, according to Theorem 2.6, 
has only one eigenvalue and one-dimensional eigensapce. In addition, according to the direct sum decomposition of the 
vector space, there shall be
, which means 
is nilpotent on cyclic space
. In what follows, we will show step by step that in this situation, there is a concise relationship between the corresponding power kernels and power images of linear map
. For simplicity, the following abbreviations of notations are introduced:
where 
means the restricted eigenspace in subspace
, and 
means the restricted image of 
on
. In what follows, the power 1 will be omitted when
, i.e.
, 
, 
,
. Then it is easy to verify that
, 
,
. The above analysis also readily yields that
,
.
Now, it is ready to present some further results for a cyclic space as follows.
Lemma 3.1 If
, then
,
.
Proof Note that
, 
are all 
subspaces, then since
, 
must have an eigenvector in
, which implies
,
. Recall that 
and
, it leads straightforward to
. Note
, then we have
.
Lemma 3.2 If
, then
,
.
Proof 
implies
, then from Lemma 3.1 there will be
,
. Let
, then it follows
which results in
, i.e.
.
From Lemma 3.2, one can see that for a cyclic subspace, the sequence of the dimensions of power images 
decrease with step one, and hence, the sequence of dimensions of power kernels 
will increase with step one.
Corollary 3.3
.
Proof Note that
, there must be
. By Lemma 3.2, it follows that
. Since
, then we have
.
Theorem 3.4 (Cross-Identity Theorem) The power kernels and power images of linear map 
on a cyclic space have the following Cross-Identity relationship, i.e.
Proof By Corollary 3.3, we can know readily that
,
. Moreover, according to the definition of
, 
, it is easy to verify the following inclusion sequence.
We give the proof in what follows by two steps. The first step is to prove
. Since
, so for
, there is a 
such that
, which yields
. Hence, 
, and therefore
. Because 
and 
have the same dimension, there must be
. Then we have
. Next step is to prove
,
. For
, we have
. Since
, then
. Hence, there must exist an 
such that 
with
. Since
, then
, which means there must exit an 
such that 
with
. Carrying on with this procedure, one can find an 
such that 
with 
. Now it can be deduced from above that
, i.e.
, which implies
. According to Corollary 3.3, 
, thus, we can readily contend that
.
Corollary 3.5 Suppose matrix 
has only one eigenvalue, and the corresponding eigenspace v is one dimensional, then there must exist a transformation matrix 
such that 
is a Jordan block matrix.
Proof Let matrix 
be the associated matrix of a linear map 
under a basis
. Since
, then there must be an 
such that
, i.e.
. Let
, then it is easy to verify that 
constitutes a fan basis of
. Thus, it is available to define an invertible linear transformation 
with an invertible transformation matrix T, such that
Notice that
, we have
, i.e.
. Then,
. From the above transformation we can contend that 
is a Jordan block matrix with the following form
4. Conclusion
This paper studies some special properties of cyclic subspaces. Although it is well known that the dimensions of the power kernels of a linear map on a vector space are decreasing, and the corresponding power images are increasing, i.e. 
and
, we show in this paper that for a cyclic subspace, the sequence of the dimensions of power kernels increases with step one, and the sequence of dimensions of power images decreases with step one. Besides, we find a Cross-Identity relationship between the power kernels and the power images, which is peculiar to cyclic subspaces. Another contribution of this paper is that we find a new way, by the application of the cross-identity theorem, to prove that a square matrix with only one eigenvalue and one-dimensional eigenspace is similar to a Jordan block matrix.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant No. 11172126). We also would like to thank the referees for their comments and suggestions.