Advances in Pure Mathematics, 2011, 1, 9- 15
doi:10.4236/apm.2011.12004 Published Online March 2011 (http://www.SciRP.org/journal/apm)
Copyright © 2011 SciRes. APM
On Bicomplex Representation Methods and Applications of
Matrices over Quaternionic Division Algebra*
Junliang Wu, Pingping Zhang
College of Mathematics & Statistics, Chongqing University, Chongqing, Chi na
E-mail: jlwu678@tom.com, zhpp04010248@163.com
Received January 6, 2011; revised January 20, 2011; accepted January 25, 2011
Abstract
In this paper, a series of bicomplex representation methods of quaternion division algebra is introduced. We
present a new multiplication concept of quaternion matrices, a new determinant concept, a new inverse con-
cept of quaternion matrix and a new similar matrix concept. Under the new concept system, many quaternion
algebra problems can be transformed into complex algebra problems to express and study. These concepts
can perfect the theory of [J.L. Wu, A new representation theory and some methods on quaternion division
algebra, JP Journal of Algebra, 2009, 14(2): 121-140] and unify the complex algebra and quaternion division
algebra.
Keywords: Quaternion Determinant, Product of Quaternion Matrix, Inverse of Quaternion Matrix, Similar
Quaternion Matrix, Application, Solution
1. Introduction
In recent years, the algebra problems over quaternion
division algebra have drawn the attention of mathematics
and physics researchers [1-12]. Quaternion algebra the-
ory is getting more and more important. In many fields
of applied science, such as physics, figure and pattern
recognition, spacecraft attitude control, 3-D animation,
people start to make use of quaternion algebra theory to
solve some actual problems. Therefore, it encourages
people to do further research [13-17] on quaternion alge-
bra theory and its applications.
The main obstacle in the study of quaternion algebra is
the non-commutative multiplication of quaternion. Many
important conclusions over real and complex fields are
different from ones over quaternion division algebra,
such as determinant, the trace of matrix multiplication
and solutions of quaternion equation. From the conclu-
sions on quaternion division algebra, we find it to lack
for general concepts, such as the definition of quaternion
matrix determinant. There are different definitions which
are given in [1,3,4,6,11,18] since Dieudonne firstly in-
troduced the quaternion determinant in 1943. In addition,
the inverse of quaternion matrix has not been well de-
fined so far, because it depends on other algebra con-
cepts. In the study of quaternion division algebra, people
always expect to get some relations between quaternion
division algebra and real algebra or complex algebra.
However, some conclusions on real or complex fields are
correct but not on quaternion division algebra. It makes
us to consider establishing other algebra concept system
over quaternion division algebra to unify the complex
algebra and quaternion division algebra.
Recently, Wu in [19] used real representation methods
to express quaternion matrices and established some new
concepts over quaternion division algebra. From these
definitions, we can see that they can convert quaternion
division algebra problems into real algebra problems to
reduce the complexity and abstraction which exist in all
kinds of definitions given in [1,3,6,10,11,20]. However,
as Wu in [19] mentioned, these concept system is not
suitable for complex algebra.
In this paper, based on the bicomplex form of quater-
nion matrix, we present some new concepts to quaternion
division algebra. These new concepts can perfect the
theory of Wu in [19] and unify the complex algebra and
quaternion division algebra.
This paper is organized as follows. In Section 2, we
introduce a complex representation method of quaternion
matrices and explore the relation between quaternion
matrices and complex matrices. In Section 3, we present
*This work was supported by National Natural Science Foundation o
f
China (No. 70872123) and Science Research Foundation of Chongqing
city of China (09-03-029).
J. L. WU ET AL.
Copyright © 2011 SciRes. APM
10
a series of new concepts over quaternion division algebra
and study their properties. In section 4, we establish
some important theorems to illustrate the applications
and effectiveness of the new concept system.
Let C denote the complex field,
H
denote the
quaternion set, mn
C denote the set of mn complex
matrices, mn
H
denote the set of mn quaternion
matrices and T
A
denote the transpose matrix of
A
.
2. The Bicomplex Representation Methods
of Quaternion Matrices and the Relation
between Quaternion Matrices and
Complex Matrices
For any quaternion matrix mn
A
H,
A
can be uniquely
represented as
01
jAA A, (2.1)
where

0, 1
mn
ss
AC , 1jA means to multiply
each entries of 1
A
by
from right hand side.
For above reasons, we can establish a mapping rela-
tion between quaternion matrices and complex matrices
as follows:s
01
:|,
mn
f
AH AA, (2.2)
where

0, 1
mn
ss
AC .
The set of mn quaternion matrices is written as
A
and the set of image of
A
is written as img
A.
Theorem 2.1. Let

01
:|,
mn
f
AH AA,


0, 1
mn
ss
AC . Then the mapping
f
is a bijec-
tive mapping from
A
to img
A.
Proof. For any entry

01
,
A
A in img
A, there exists
the corresponding quaternion matrix 01
jAA A in
A
, therefore
f
is a surjection from
A
to img
A. Si-
multaneously, since any quaternion matrix in
A
can be
uniquely represented as the form (2.1), so
f
is an in-
jection from
A
to img
A. Thus
f
is a bijective map-
ping from
A
to img
A.
The proof is complete.
Theorem 2.2. Bijection

01
:,f
A
AA,

0,1
mn
s
AC s
is an isomorphism mapping from
A
to img
A.
By the concept of isomorphism mapping, this theorem
is easy to prove and we omit it here.
We shall mention that Theorem 2.2 is the foundation
of this article, because isomorphism vector spaces have
the same properties.
3. The Bicomplex Matrix Concept System
over Quaternion Division Algebra
According to the complex representation of quaternion
matrices above, a series of new definitions of quaternion
division algebra which are helpful to discuss the algebra
problems on quaternion division algebra can be given as
follows.
Definition 3.1. The matrix j
E
EE
is said to be
a nn
unit quaternion matrix if E is a nn
unit
matrix over complex field. In particular, if 1n
, then
jj
11
EEE is said to be a unit quaternion writ-
ten as u
a.
Definition 3.2. Let and 01
nt
j
 BB BH be given.
The operator 00 11
j
ABABAB (where 00 11
,AB AB
are both the multiplications of complex matrices) is
called the *-product of quaternion matrices A and B. In
particular, if 1mnt
, then we can derive the
*-product of quaternions.
Note: when ,
mn nt
ACBC , then 
A
BAB.
Under the Definition 3.1 and Definition 3.2, we give
some relative properties.
For any matrix ,nn
AB H, we have:
1)


E
AAEA, where
E
is a nn
unit
quaternion matrix;
2)

A
BBA;
3)

A
BCACBC;
4)

TTT
ABB A;
5)
Tr Tr

A
BBA.
Similarly, we establish a new definition as follows.
Definition 3.3. Let 1n
X
H and a
H
be given. Then
00 11
aaaaj
 
X
XXX is called the *-product of
quaternion and quaternion vector, where 01
j
X
XX,
11
01
,
nn
CCXX
, 01
aa aj
, 01
,aCaC.
Now, we introduce the following concept to quater-
nion division algebra.
Definition 3.4. For any quaternion matrix nn
A
H
01
j
A
AA, 01
j
A
AA is said to be the de-
terminant of
A
, where . is the determinant of a com-
plex matrix.
Note: when nn
AC , then
A
A.
The Definition 3.3 is reasonable. First of all, the result
of a quaternion matrix determinant under Definition3.4 is
a quaternion. Secondly, from Definition 3.4 we can see
that it can convert the determinant of a quaternion matrix
into that of complex matrices to reduce the complexity
and abstraction. Finally, the new determinant has the
same fundamental properties as that over complex field.
That is, if
A
is a nn
quaternion matrix andij
,
then we have
1) T
AA.
2) If quaternion matrix B is obtained from quater-
nion matrix
A
by interchanging two rows (or columns)
of
A
, then BA.
J. L. WU ET AL.
Copyright © 2011 SciRes. APM
11
3) If quaternion matrix
A
has a zero row (or column),
then 0
T
AA .
4) n
kk
AA, wheren
n
kkk k
 
 , k
H
.
5) If the jth row (column) of quaternion matrix
A
equal a multiple of the ith row (column) of the matrix,
then 0A.
6) Suppose that
A
, B and C are all nn
qua-
ternion matrices. If all rows of B and C both equal
the corresponding to rows (columns) of
A
except that
the ith row (column) of
A
equal the sum of the ith of
B and C, then 
A
BC.
7) If quaternion matrix B is the nn matrix re-
sulting from adding a multiple of the ith row (or column)
of matrix
A
to the jth row (or column) of matrix
A
,
then BA.
8) Let
A
and B be nn quaternion matrices re-
spectively. We have  
A
BAB.
Up to now, people still treat the inverse matrix concept
of quaternion matrix as complex matrix, that is, if qua-
ternion matrix
A
satisfies 1
A
AE (where
E
is a
real unit matrix), then people think that quaternion ma-
trix
A
exists its inverse matrix 1
A
. However, people
pointedly ignore two questions. An issue is how to define
the product of quaternion matrices 1
A
and
A
. The
other one is how to make a calculation of 1
A
.
It indicates that the terminology of inverse matrix does
not have a clear definition in quaternion algebra theory.
In the following, we shall give a new definition and
specific computational method for the inverse of quater-
nion matrix.
Definition 3.5. Let 01
nn
j
 AA AH be given
(where 01
, AA both are complex matrices). If the inverse
matrices of 0
A and 1
1
A
both exist, then quaternion
matrix
A
is said to be invertible and the inverse matrix
is written as 11
01
j
 
AAA, where 1
0
A
, 1
1
A
denote
the inverse of complex matrices 0
A, 1
A
respectively.
Note: when nn
AC , then 1
A
A.
The inverse of quaternion matrix under the new defi-
nition has the same fundamental properties as those un-
der the traditional algebra system. It is easy to show the
following facts by the new concept, namely, if a quater-
nion matrix
A
is invertible, then we have:
1)

AA.
2)

11
01
kkk
kj

 AAAA, where
k
A
AA A
is product of kA which is defined
in Definition 3.2.
3) If 12
,,,
m
AA A are all invertible quaternion ma-
trices, then

121 1mmm

 AAAA AA.
Obviously, by the new definition of inverse of quater-
nion matrix above, people can determine easily whether
the inverse matrix of quaternion matrix exists or not and
calculate the inverse matrix if possible.
Under the definition of inverse of quaternion matrix
above, a new concept of similar quaternion matrices can
be given as follows:
Definition 3.6. Let ,nn
AB H, if there exists an
invertible quaternion matrix P such that

A
PBP,
then
A
and B are said to be similar quaternion ma-
trices written as
A
B.
Note: when ,nn
AB C,

A
PBP is equiva-
lent to 1
00
APBP, where 01
PPP
j
, 01
,nn
P
PC.
For similar quaternion matrices, we will deduce many
important properties in the next section.
4. Some Applications of the Bicomplex
Matrix Concept System
In this section, we establish some important theorems to
illustrate the applications and effectiveness of the new
concept system for the research of quaternion division
algebra. The eigenvalue is an important issue in quater-
nion division algebra theory, so under the new concept
system, we will study firstly the eigenvalues of quater-
nion matrix and the relation between eigenvalues of
similar quaternion matrices in detail.
Before showing the application, we’ll introduce firstly
some concepts associated with eigenvalue.
Definition 4.1. For any matrix

nn
ij
a
HA , if
there exists nonzero quaternion vector 1n
X
H and a
quaternion 01
j

(where 0
, 1
are both com-
plex numbers) such that
A
X
X, then
is said
to be the left eigenvalue of
A
, and
X
is the left ei-
genvector corresponding to
.
For the sake of distinction, we call the left eigenvalue
and the left eigenvector under Definition 4.1 the left
quaternion eigenvalue and the left quaternion eigenvec-
tor respectively.
According to the new definition of quaternion matrix
multiplication and
A
X
X, we can derive that
0
EAX . Thus

f


E
A is said
to be the characteristic polynomial of
A
(where the op-
erator
denotes the determinant of quaternion matrix
under Definition 3.4).
Theorem 4.1. A nn
quaternion matrix
01
j
AA A (where0
A, 1
A both are complex matri-
ces), if
and
are the left eigenvalues of 0
A and
1
A respectively, then aj
andbj
(aC
,
bC
) are the left quaternion eigenvalues of
A
.
Proof. Since
and
are the left eigenvalues of
0
A and 1
A respectively, then there exist nonzero vectors
J. L. WU ET AL.
Copyright © 2011 SciRes. APM
12
1n
C
and 1n
C
such that 01
,
AA
 
.
We have

 
01 0
jj aj

 0AAA A
 
,
for aC .




01 1
jjjjj
bj j
 
 
0AAA A

,
for bC .
So aj
and bj
are all the left quaternion ei-
genvalues of
A
.
The proof is complete.
Similarly, we introduce a new right quaternion eigen-
value concept.
Definition 4.2. For any matrix

nn
ij
a
HA , if
there exists nonzero quaternion vector 1n
H
and qua-
ternion 01
j

 (where01
,
are both complex
numbers) such that
YA Y, then
is said to be the right quater-
nion eigenvalue of
A
, and Y is the right quaternion
eigenvector corresponding to
.
For the right eigenvalue of quaternion matrix, we have
the following theorem:
Theorem 4.2. A nnquaternion matrix 01
jAA A
(where 0
A, 1
Aare both complex matrices), if
and
are the right eigenvalues of 0
A an 1
A respectively,
then aj
and bj
(,aCbC ) are the right
quaternion eigenvalues of
A
.
Proof. Since
and
are the right eigenvalues of
0
A and 1
A respectively, then there exist nonzero vectors
1n
C
and 1n
C
such that 01
,
AA
 
.
We have



01 0
, for
jj
aja C

 
 
0
 

AAAA
.
 


01 1
,for
jjjjj
bj jbC
 
 
0

AAAA
.
So aj
and bj
are the right quaternion ei-
genvalues of
A
.
The proof is complete.
Theorem 4.3. If the left eigenvalues of 0
A are
12
,,,
k

and the left eigenvalues of 1
A are
12
,,,
m

(where 0
A,1
A both are complex matri-
ces), then the left quaternion eigenvalues of matrix
01
jAA A are
saj
or
, , , 1,,, 1,,
t
bjaCbCs ktm
.
Proof. Suppose that
is arbitrary left quaternion
eigenvalue of
A
, then 0
, 1
01
n
j
 
H

,
such that
A
, that is, 00 00
11 11
A
A


. Since
0
, we know that both 0
and 1
are not zeroes.
So there are two cases as follows:
1) When
0
, obviously, we have
012
,,,
k
 
.
So,
,1,2,,
iaj ik

 .
2) When
0
, obviously, we have
112
,,,
m
 
.
So,
,1,2,,
t
bjtm

 .
To sum up 1), 2) and Theorem 4.1, we can draw the
conclusion.
The proof is complete.
Theorem 4.4. If the right eigenvalues of 0
A are
12
,,,
k

and the right eigenvalues of 1
A are
12
,,,
m

(where 0
A,1
A both are complex matri-
ces), then the right quaternion eigenvalues of matrix
01
j
AA A are
saj
or
, , , 1,,, 1,,
t
baCbCsktm
.
This proof is similar toTheorem 4.3. So we omit it
here.
Theorem 4.5. Let nn
A
H
, then
A
and T
A
have
the same quaternion left (right) eigenvalues.
Proof. Since 01
j
AA A (where
0
nn
A
C,1
nn
C
A), then 01
TTT
j
A
AA
. We know
i
A and T
i
A
have the same left (right) eigenvalues
(1, 2i
). By Theorem 4.3 and Theorem 4.4, we can
draw the conclusion.
The proof is complete.
Theorem 4.6. Let nn
A
H
and ,
H
be
given. If
is the left (right) quaternion eigenvalue
of
A
, then
is the right (left) quaternion eigen-
value of
A
.
Proof. Since
is the left quaternion eigenvalue of
A
, then there exits nonzero vector
such that
A
. Then

TT
A

, we can have
TT T
A
. So
is the right quaternion eigen-
value of T
A
, by Theorem 4.5, we know
is the right
quaternion eigenvalue of
A
. The same proof to
.
So, the proof is complete.
Specially, when nn
AC , if

is the left (right)
eigenvalue of
A
, then
is the right (left) eigen-
value of
A
.
Note: By the new definition of quaternion multiplica-
tion, the left quaternion eigenvalue of a quaternion ma-
trix is equivalent to its right quaternion eigenvalue. So
they are both called quaternion eigenvalue of the quater-
nion matrix.
In the following, we show an important result.
Theorem 4.7. Let ,nn
AB H be given. If
A
B,
then
A
and B have the same eigenvalues.
Proof. Since
A
B, there exists an invertible matrix
nn
PH such that

A
PBP, that is equivalent to
1
00
APBP and 1
1111
A
BPP (where 01
jAA A,
01
j
B
BB, 01
j
PPP). We know
s
B
and
s
A
0,1s have the same eigenvalues. By Theorem 4.3
and Theorem 4.4, we can draw that
A
and B have
J. L. WU ET AL.
Copyright © 2011 SciRes. APM
13
the same eigenvalues.
The proof is complete.
Theorem 4.8 (The generalized Cayley-Hamilton theo-
rem over quaternion division algebra). A quaternion ma-
trix
A
must be the root of its characteristic polynomial

f


E
A.
Proof. According to Definition 3.4, we know that:





01
01 01
00110 1
ff j
jj
jgh j


 



EA
EE AA
EA EA
,
where

00 0
g


E
A,
11 1
h


E
A.
According to the Cayley-Hamilton theorem on com-
plex field, we know

0
g0A,

1
h0A. So,



01
fgh0AAA
. It indicates that quaternion
matrix
A
must be the root of its characteristic polyno-
mial

f
.
So, the proof is complete.
Theorem 4.9. Let (where 01
nn
j
 AA AH
01
,nn
A
AC) be given.
A
is a diagonalizable matrix
if and only if both 0
A and 1
A are diagonalizable ma-
trices.
Proof.
A
is diagonalizable matrix , that is ,there ex-
its an invertible quaternion matrix P such that

A
PP. It is equivalent to 1
0000
AP P and
1
111
AP P (where 01
j is diagonal matrix).
So,
A
is diagonalizable matrix if and only if both 0
A
and 1
A are diagonalizable matrices.
The proof is complete.
Corollary 4.9. Let 01
nn
j
 AA AH (where
01
,nn
A
AC) be given. If 0
A and 1
A both have n
different eigenvalues, then
A
is diagonalizable matrix.
Corollary 4.9'. Let 01
nn
j
 AA AH (where
01
,nn
A
AC) be given. Quaternion matrix
A
is di-
agonalizable matrix if and only if and 1
A both have n
linearly independent eigenvactors.
Corollary 4.9''. Let 01
nn
j
 AA AH (where
01
,nn
A
AC) be given. Quaternion matrix
A
is di-
agonalizable matrix if and only if the geometric multi-
plicity of 0
A and 1
A both equal their algebraic multi-
plicity respectively.
In Section 3, we have given the new definition of the
inverse of quaternion matrix, but that of quaternion is not
defined. In fact, a quaternion can be treated as a 11
matrix. So we can define the inverse of quaternion as
follows:
Definition 4.3. For any quaternion 01
aa aj, if
neither of 0
a and 1
a are zeroes, then 11
01
aaaj
 

is said to be the inverse of a, where

10,1
s
as
is
the reciprocal of
s
a.
It is easy to verify the following facts. For any
,ab H
, we have:
1) uu
aaaa a
 ;
2) abba
;
3)
ab cacbc
;
4)


01
nn
n
aa aj ;
5) If 01
aa aj
has the inverse a, then u
aa a
.
In addition, we discover that there are some special
phenomena about the roots of quaternion polynomial
under the new definition of quaternion multiplication.
Definition 4.4. The polynomial which has the form as
follows:

*1
*0*1 *0
01 1
nnn
axaxax ax
  is
said to be quaternion polynomial with complex coeffi-
cients (where , 0,1,,
i
ai n
are all complex numbers,
01
x
xxj
, *0
x
is the *-product of iquaternion
x
and 0
x
is unit quaternion).
Theorem 4.10. Let
f
x be a quaternion polyno-
mial with complex coefficients. Then

f
x has infinite
quaternion roots.
Proof. By Fundamental Theorem of algebra,
f
x
exists at least one complex root 0
x
, then for any given
complex number 1
x
, obviously, 01
x
xj is the root of
f
x.
The proof is complete.
Theorem 4.11. Let
f
x be a quaternion polyno-
mial with complex coefficients and 01
nn
j
 AA AH
be a given quaternion matrix (where, both 0
A and 1
A
are compex matrices). If
is the eigenvalue of 0
A,
then
f
is the eigenvalue of

f
A
.
Proof. According to the new definition of quaternion
multiplication, we can easily obtain

0
ff
A
A.
Since
is the eigenvalue of 0
A, so

f
is the ei-
genvalue of
0
f
A
.
The proof is complete.
Under the new concept system, we can also solve the
problems of existence and uniqueness of the solutions to
the quaternion system of linear equations
A
X
b,
where operator ‘
’ denotes the new multiplication of
quaternion matrices.
As we known, for any mn
A
H
,
A
can be repre-
sented uniquely as 01
j
AA A, where
s
A
0,1s
are nn
complex matrices. Let

T
1011202101
,,,
nn
x
xjx xjx xj X and

T
1011 202101
,,,
nn
bbjbbj bbj b be 1n
quater-
nion vectors, then the following theorems are valid.
Theorem 4.12. Let 01
nn
j
 AA AH be given
and 01
j
X
XX be 1n
quaternion vector. If rank
s
s
r
A and the fundamental system of solutions to the
system of homogeneous linear equations ss
0AX is

1,2 ,, i
ii in r

0,1s respectively, then any solu-
tion to the quaternion system of homogeneous linear
J. L. WU ET AL.
Copyright © 2011 SciRes. APM
14
equations 0A
X
can be expressed as follows:




00
11
01 01020200
11 11121211
nrnr
nr nr
cc c
cc cj
 
 




X
,
where , 0,,, 0,1
s
st ss
cCtnrs .
Proof. By the new definition of quaternion matrix
multiplication, the quaternion system of homogeneous
linear equations 0AX is equivalent to the system
of homogeneous linear equations 00
11
0
0
X
X
A
A. Since any
solution to the system of homogeneous linear equations
ss
0AX can expressed as


00
1122sssss sn rsn r
cc c
 

X
(where , 1,,, 0,1
s
st ss
cCtnrs )
and the solutions of the quaternion system of homoge-
neous linear equations 0A
X
are 01
j
X
XX.
So we can draw the conclusion.
So, the proof is complete.
Corolla ry 4.12. Let 01
jAA A be a given quater-
nion matrix (where mn
s
AC
, 0,1s).
If rank

0
A
= rank

1
A
= n, then the quaternion
system of homogeneous linear equations 0A
X
has
unique solution

T
0, 0,00X.
Corollary 4.12'. Let 01
jAA A be a given qua-
ternion matrix (wheremn
s
AC ,0,1s). If rank

0nA and rank

1nA, then the quaternion sys-
tem of homogeneous linear equations 0A
X
only
exists complex solutions.
Theorem 4.13. Let 01
jAA A be a given quater-
nion matrix, 01
j
X
XX and 01
jbb b be qua-
ternion vectors (where mn
s
AC, 1n
s
XC,

T
12
,,,
ssssn
bb bb, st
bC, 0,1s, 1, 2,,tn).
If there is at least one
00,1s such that rank
0
s
A
rank

00
ss
A
b, then the quaternion system of linear
equations A
X
b has no solution.
Proof. By the new definition of quaternion matrix
multiplication, the quaternion system of linear equations
A
X
b is equivalent to the system of linear equa-
tions 00 0
11 1
b
X
b
A
A
X, since
 
000
rank ra<nk
sss
A
Ab, the
system of linear equations 00 0
11 1
b
X
b
A
A
X have no solution,
that is, the quaternion system of linear equations
A
X
b has no solution.
So, the proof is complete.
Theorem 4.14. Let 01
jAA A be a given quater-
nion matrix and 01
j
X
XX be a given quaternion
vector (where mn
s
AC, 1n
s
XC, 0,1s). We sup-
pose that the fundamental system of solutions to the sys-
tem of linear equations ss
0AX is

1,2 ,,
s
ss
s
nr

0,1s respectively and

0,1
ss
is a special so-
lution of the system of linear equations
s
ss
A
Xb
re-
spectively, and rank
s
A
= rank

s
s
Ab
0, 1s,
then any solution to the quaternion system of linear
equations
A
X
b can be expressed as:



00
11
001010202 00
111111212 11
nr nr
nr nr
cc c
cc cj
 
 


 
 
X
.
Proof. By the new definition of quaternion matrix
multiplication, the quaternion system of linear equations
A
X
b is equivalent to the system of linear equa-
tions 00 0
11 1
b
X
b
A
A
X. Since any solution to the system of
linear equations
s
ss
A
X
b can expressed as follows:

1122
s
s
ssssss
s
nr snr
cc c


 X (where
, 1,2,,, 0,1
st s
cCtnrs
), so any solution to the
system of quaternion linear equations AX b can be
expressed as:



00
11
001010202 00
111111212 11
nr nr
nr nr
cc c
cc cj
 
 


 
 
X
.
The proof is complete.
Theorem 4.15. Let 01
j
AA A be a given quater-
nion matrix, 01
j
X
XX and 01
jbb b be quater-
nion vectors (where mn
s
AC, 1n
s
XC, 1n
sC
b,
0,1s). If rank
s
A
= rank

s
s
Ab = n
0,1s,
then the quaternion system of linear equations
A
X
b
exists unique solution.
Proof. By the new definition of quaternion matrix
multiplication, the quaternion system of linear equations
A
Xb
is equivalent to the system of linear equations
00 0
11 1
b
X
b
A
A
X, and rank
s
A
= rank

s
s
Ab = n, we
know the system of linear equations 00 0
11 1
b
X
b
A
A
X have
unique solution. So the quaternion system of linear equa-
tions
A
X
b exists unique solution.
The proof is complete.
Corollary 4.15. Let 01
j
AA A be a given nn
quaternion matrix and 01
j
bb b be a given 1n
quaternion vector. If rank

s
A
= rank

s
s
Ab = n
0,1s, then the solution of the quaternion system of
equations
A
X
b is 1
XA b.
Corollary 4.15'. Let mn
AC and 01
j
bb b
(where 1
1
, 0,1,
n
ss
0bCb ) be given. Then the
J. L. WU ET AL.
Copyright © 2011 SciRes. APM
15
quaternion system of linear equations A
X
b has no
solution.
Corollary 4.15''. Let mn
AC and 1n
bC be
given. If rank

A
= rank

Ab , then any solution to the
quaternion system of linear equations A
X
b can
expressed as 1aj
AXb, where aC.
5. References
[1] L. X. Chen, “Inverse Matrix and Properties of Double
Determinant over Quaternion TH Field,” Science in
China (Series A), Vol. 34, No. 5, 1991, pp. 25-35.
[2] L. X. Chen, “Generalization of Cayley-Hamilton Theo-
rem over Quaternion Field,” Chinese Science Bulletin,
Vol. 17, No. 6, 1991, pp. 1291-1293.
[3] R, M. Hou, X. Q. Zhao and l. T. Wang, “The Double
Determinant of Vandermonde’s Type over Quaternion
Field,” Applied Mathematics and Mechanics, Vol. 20, No.
9, 1999, pp. 100-107.
[4] L. P. Huang, “The Determinants of Quateruion Matrices
and Their Propoties,” Journal of Mathematics Study, Vol.
2, 1995, pp. 1-13.
[5] J. L. Wu, L. M. Zou, X. P. Chen and S. J. Li, “The Esti-
mation of Eigenvalues of Sum, Difference, and Tensor
Product of Matrices over Quaternion Division Algebra,”
Linear Algebra and its Applications, Vol. 428, 2008, pp.
3023-3033. doi:10.1016/j.laa.2008.02.008
[6] T. S. Li, “Properties of Double Determinant over Quater-
nion Field,” Journal of Central China Normal University,
Vol. 1, 1995, 3-7. doi:10.1007/BF02652076
[7] B. X. Tu, “Dieudonne Determinants of Matrices over a
Division Ring,” Journal of Fudan university, 1990A, Vol.
1, pp. 131-138.
[8] B. X. Tu, “Weak Direct Products and Weak Circular
Product of Matrices over the Real Quaternion Division
Ring,” Journal of Fudan University, Vol. 3, 1991, p. 337.
[9] J. L. Wu, “Distribution and Estimation for Eigenvalues of
Real Quaternion Matrices,” Computers and Mathematics
with Applications, Vol. 55, 2008, pp. 1998-2004.
doi:10.1016/j.camwa.2007.07.013
[10] B. J. Xie, “Theorem and Application of Determinants
Spread out of Self-Conjugated Matrix,” Acta Mathe-
matica Sinica, Vol. 5, 1980, pp. 678-683.
[11] Q. C. Zhang, “Properties of Double Determinant over the
Quaternion Field and Its Applications,” Acta Mathe-
matica Sinica, Vol. 38, No. 2, 1995, pp. 253-259.
[12] W. J. Zhuang, “Inequalities of Eigenvalues and Singular
Values for Quaternion Matrices,” Advances in Mathe-
matics, Vol. 4, 1988, pp. 403-406.
[13] W. Boehm, “On Cubics: A Survey, Computer Graphics
and Image Processing,” Vol. 19, 1982, pp. 201-226.
doi:10.1016/0146-664X(82)90009-0
[14] G. Farin, “Curves and Surfaces for Computer Aided
Geometric Design,” Academic Press, Inc., San Diego CA,
1990.
[15] K. Shoemake, “Animating Rotation with Quaternion
Calculus,” ACM SIGGRAPH, 1987, Course Notes, Com-
puter Animation: 3–D Motion, Specification, and Con-
trol.
[16] Q. G. Wang, “Quaternion Transformation and Its Appli-
cation to the Displacement Analysis of Spatial Mecha-
nisms, Acta Mathematica Sinica, Vol. 15, No. 1, 1983, pp.
54-61.
[17] G. S. Zhang, “Commutativity of Composition for Finite
Rotation of a Rigid Body,” Acta Mechanica Sinica, Vol.
4, 1982.
[18] E. T. Browne, “The Characteristic Roots of a Matrix,”
Bulletin of the American Mathematical Society, Vol. 36,
1930, pp. 705-710.
doi:10.1090/S0002-9904-1930-05041-7
[19] J. L. Wu and Y. Wang, “A New Representation Theory
and Some Methods on Quaternion Division Algebra,”
Journal of Algebra, Vol. 14, No. 2, 2009, pp. 121-140.
[20] Q. W. Wang, “The General Solution to a System of Real
Quaternion Matrix Equation,” Computer and Mathemat-
ics with Applications, Vol. 49, 2005, pp. 665-675.
doi:10.1016/j.camwa.2004.12.002
[21] G. B. Price, “An Introduction to Multicomplex Spaces
and Functions,” Marcel Dekker, New York, 1991.
[22] D. Rochon, “A Bicomplex Riemann Zeta Function,” To-
kyo Journal of Mathematics, Vol. 27, No. 2, 2004, pp.
357-369.
[23] S. P. Goyal and G. Ritu, “The Bicomplex Hurwitz Zeta
function,” The South East Asian Journal of Mathematics
and Mathematical Sciences, 2006.
[24] S. P. Goyal, T. Mathur and G. Ritu, “Bicomplex Gamma
and Beta Function,” Journal of Raj. Academy Physical
Sciences, Vol. 5, No. 1, 2006, pp. 131-142.
[25] J. N. Fan, “Determinants and Multiplicative Functionals
on Quaternion Matrices,” Linear Algebra and Its Appli-
cations, Vol. 369, 2003, pp. 193-201.
doi:10.1016/S0024-3795(02)00722-X
[26] Q. W. Wang, “A System of Four Matrix Equations over
Von Neumann Regular Rings and It Applications,” Acta
Mathematica Sinica, Vol. 21, 2005, pp. 323-334.
doi:10.1007/s10114-004-0493-1
[27] Q. W. Wang, “A System of Matrix Equation and a Linear
Matrix Equation over Arbitrary Regular Rings with Iden-
tity,” Applied Linear Algebra, Vol. 384, 2004, pp. 43-54.
doi:10.1016/j.laa.2003.12.039
[28] W. J. Zhuang, “The Guide of Matrix Theory over Qua-
ternion Field,” Science Press, Beijing, 2006, pp. 1-50.
[29] W. L. LI, “Quaternion Matrices,” “National Defense
Science and Technology University,” Vol. 6, 2002, pp.
73-74.
[30] R. X. Jiang, “Linear Algebra,” People’s Educational
Press, China, 1979, pp. 41-42.