Adv. Appl. Clifford Algebras 24 (2014), 769–779
© 2014 Springer Basel
0188-7009/030769-11
published online February 8, 2014
DOI 10.1007/s00006-014-0449-1
Advances in
Applied Clifford Algebras
Commutative Quaternion Matrices
Hidayet Hüda Kösal∗ and Murat Tosun
Abstract. In this study, we introduce the concept of commutative quaternions and commutative quaternion matrices. Firstly, we give some
properties of commutative quaternions and their fundamental matrices.
After that we investigate commutative quaternion matrices using properties of complex matrices. Then we define the complex adjoint matrix
of commutative quaternion matrices and give some of their properties.
Keywords. Commutative quaternions, fundamental matrices, commutative quaternion matrices.
1. Introduction
Real quaternions were introduced by Hamilton (1805-1865) in 1843 as he
looked for ways of extending complex numbers to higher spatial dimensions.
So the set of real quaternions can be represented as [5]
Q = {q = t + xi + yj + zk : t, x, y, z ∈ R and i, j, k ∈
/ R}
where
i2 = j 2 = k 2 = −1, ij = −ji = k, jk = −kj = i, ki = −ik = j.
From these rules it follows immediately that multiplication of quaternions
is not commutative. So, it is not easy to study quaternion algebra problems. Similary, it is well known that the main obstacle in study of quaternion
matrices, dating back 1936 [11], is the non-commutative multiplication of
quaternions. There are a lot of works associated with quaternion matrices.
Zhang has given a brief survey on quaternions and matrix of quaternions
[12]. Because of noncommutativity of quaternions, the theory eigenvalues of
a quaternion matrix becomes one of the special topics about quaternion matrices [7]. So, Baker has discussed right eigenvalues for quaternionic matrices
with a topological approach in [1]. On the other hand, Huang has introduced on left eigenvalues of quaternionic matrix [6]. And some differences
between right and left eigenvalues of quaternion matrices has been given in
∗ Corresponding
author.
770
H.H. Kösal and M. Tosun
Adv. Appl. Clifford Algebras
[13], including the Gershgorin type theorems for right and left eigenvalues of
quaternionic matrices. As a continuation of [13], Farid, Wang and Zhang have
discussed some differences of eigenvalue problem of complex and quaternionic
matrices in [4].
After the discovery of quaternions by Hamilton, Segre proposed modified
quaternions so that commutative property in multiplication is possible [10].
This number system, called commutative quaternion. Commutative quaternions are decomposable into two complex variables [2]. The set of commutative quaternions is 4− dimensional like the set of quaternions. But this set
contains zero-divisor and isotropic elements. As has been noticed, there is no
theory of commutative quaternion matrices. However, commutative quaternion algebra theory is becoming more and more important in recent years
and has many important applications to areas of mathematics and physics
[3, 8, 9].
The present article, we will investigate commutative quaternion matrices. Firstly, we will give some properties of commutative quaternions which
will be used in the following sections. After that, we will introduce commutative quaternion matrices and give some properties of them. Finally, we define
the complex adjoint matrix of commutative quaternion matrices and we give
some properties of these matrices. Then we will define the determinant of a
commutative quaternion matrices using its adjoint matrix.
2. Commutative Quaternions
A set of commutative quaternions are denoted by [2]
H = {q = t + ix + jy + kz : t, x, y, z ∈ R and i, j, k ∈
/ R}
where the basis elements i, j, k satisfy the following multiplication rules:
i2 = k 2 = −1, j 2 = 1, ijk = −1, ij = ji = k, jk = kj = i, ki = ik = −j.
For q = t + ix + jy + kz , there exists three kinds of its conjugate, called
principal conjugations:
q (1) = t − ix + jy − kz,
q (2) = t + ix − jy − kz,
q (3) = t − ix − jy + kz.
Principal conjugations have the following properties:
t=
y=
q+q (1) +q (2) +q (3)
,
4
q+q (1) −q (2) −q (3)
j
,
4
+q (2) −q (3)
−4
(1)
(2)
+q (3)
k q−q −q
.
−4
x = i q−q
z=
(1)
Definition 2.1. The norm of q = t + ix + jy + kz is defined as
4
q = q q (1) q (2) q (3)
2
= (t + y) + (x + z)
2
2
(t − y) + (x − z)
2
≥ 0.
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Commutative Quaternion Matrices
771
Scalar and vector parts commutative quaternion q are denoted by Sq = t and
−
→
Vq = ix + jy + kz, respectively.
Multiplication of any commutative quaternions q = t + ix + jy + kz and
q1 = t1 + ix1 + jy1 + kz1 are defined in the following ways,
qq1 = tt1 − xx1 + yy1 − zz1 + i (xt1 + tx1 + zy1 + yz1 )
+j (ty1 + yt1 − xz1 − zx1 ) + k (zt1 + tz1 + xy1 + yx1 ) .
⎛
⎞
t
⎜ x ⎟
⎟
We will denote any commutative quaternion q ∼
=⎜
⎝ y ⎠ . Then multiplicaz
tion of q and q1 can be represented by an ordinary matrix-by-vector product
⎞⎛
⎛
⎞
t1
t −x y −z
⎜
⎜ x t
⎟
z y ⎟
⎟ ⎜ x1 ⎟
qq1 = q1 q ∼
= L q q1 = ⎜
⎝ y −z t −x ⎠ ⎝ y1 ⎠ ,
z y x t
z1
where Lq is called fundamental matrix of q.
Theorem 2.2. If q and q1 are commutative quaternions and α, β are real
numbers, then the following identities hold:
1. q = q1 ⇔ Lq = Lq1 ,
2. Lq+q1 = Lq + Lq1 , Lqq1 = Lq Lq1 ,
3. Lαq+βq1 = αLq + βLq1 ,
4
4. T r (Lq ) = q + q (1) + q (2) + q (3) = 4t, q = det (Lq ) .
Theorem (2.2) can be proved by simple matrix computation.
Theorem 2.3. Every commutative quaternion can be represented by a 2 × 2
complex matrix.
Proof. Let q = t + ix + jy + kz ∈ H, then every commutative quaternion can
be uniquely expressed as q = c1 + jc2 , where c1 = t + ix, c2 = y + iz ∈ C (C
is the set of complex numbers). The linear map ϕq : H → H is defined by
ϕq (p) = qp for all p ∈ H. This map is bijective and
ϕq (1) = c1 + jc2
ϕq (j) = c2 + jc1 ;
with this transformation, commutative quaternions are defined as subset of
the matrix ring M2 (C) , the set of 2 × 2 complex matrices:
c1 c2
N=
: c 1 , c2 ∈ C .
c2 c1
Then, H and N are essentially the same.
Note that
ϕ:H→N
q = c1 + jc2 → ϕ (q) =
c1
c2
c2
c1
4
is bijective and preserves the operations. Furthermore q = |det ϕ (q)| .
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H.H. Kösal and M. Tosun
Adv. Appl. Clifford Algebras
3. Commutative Quaternion Matrices
The set Mm×n (H) denotes all m×n type matrices with commutative quaternion entries. If m = n, then the set of commutative quaternion matrices is denoted by Mn (H) . For A = (aij ) ∈ Mm×n (H) , B = (bij ) ∈ Mm×n (H) and
C = (cjk ) ∈ Mn×p (H) , the ordinary matrix addition and multiplication are
defined by
⎞
⎛
n
A + B = (aij + bij ) ∈ Mm×n (H) , AC = ⎝
aij cjk ⎠ ∈ Mm×p (H) .
j=1
The scalar multiplication is defined as, for q ∈ H,
qA = Aq = (qaij ) .
It is easy to see that for A, C
(qA) C = q (AC) ,
(Aq) C = A (qC) ,
(qq1 ) A = q (q1 A) .
Moreover, Mm×n (H) is free module over the ring H. For A ∈ Mm×n (H) ,
there exists three kinds of its conjugates, called principal conjugates:
(1)
(2)
A(1) = aij ∈ Mm×n (H) , A(2) = aij ∈ Mm×n (H) and
(3)
A(3) = aij ∈ Mm×n (H) .
T
AT = (bji ) ∈ Mm×n (H) is the transpose of A ; A†i = A(i) ∈ Mm×n (H)
is the ith conjugate transpose of A.
A square matrix A ∈ Mn (H) is said to be normal matrix by ith conjugate, if
AA†i = A†i A; Hermitian matrix by ith conjugate, if A = A†i ; unitary matrix
by ith conjugate , if AA†i = I where I is the identity matrix and invertible
matrix, if AB = BA = I for some B ∈ Mn (H) .
Theorem 3.1. Let A ∈ Mm×n (H) , B ∈ Mn×p (H) . Then the following properties hold:
T (i)
;
I. A(i) = AT
†
II. (AB) i = B †i A†i ;
T
III. (AB) = B T AT ;
(i)
IV. (AB) = A(i) B (i) ;
−1
V. (AB) = B −1 A−1 if A and B is invertible;
−1 −1 †i
= A
if A is invertible;
VI. A†i
(i) −1 −1 (i)
= ⎧
A
;
VII. A
(k)
A
i = j = k
⎨
(j)
VIII. A(i)
=
for i, j, k ∈ {1, 2, 3} ;
⎩
A
i=j
where A(i) is the ith conjugate and i = 1, 2, 3.
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Commutative Quaternion Matrices
773
Proof. I, III, IV, V, VI, VII and VIII can be easily shown. We will prove II.
II. Let A = A1 + jA2 and B = B1 + jB2 , where A1 , A2 , B1 and B2 are
complex matrices. Then
(AB)
†1
= [(A1 + jA2 ) (B1 + jB2 )]
†1
T
= (A1 B1 ) + (A2 B2 ) + j (A1 B2 ) + (A2 B1 )
T
= A1 B 1 + A2 B 2 + j A1 B 2 + A2 B 1
T T T T
T T T T
= B1
A1 + B 2
A2 + j B 2
A1 + B 1
A2
.
On the other hand, for
T
T
and
B †1 = B 1 + j B 2
T
T
A†1 = A1 + j A2 ,
T T T T
T T T T
B † 1 A† 1 = B 1
A1 + B 2
A2 +j B 2
A1 + B 1
A2
.
Thus, we obtain (AB)
(AB)
†1
= B † 1 A† 1 .
†2
†
= [(A1 + jA2 ) (B1 + jB2 )] 2
T
= [(A1 B1 ) + (A2 B2 ) − j [(A1 B2 ) + (A2 B
1 )]]
= B1T AT1 + B2T AT2 − j B2T AT1 + B1T AT2 .
On the other hand , for
B †2 = B1T − jB2T and
A†1 = AT1 − jAT2 ,
B †2 A†2 = B1T AT1 + B2T AT2 − j B2T AT1 + B1T AT2 .
Thus, we have (AB)
†2
= B †2 A†2 . The same way, we have (AB)
†3
= B †3 A†3 .
Theorem 3.2. Let A, B ∈ Mn (H) , if AB = I, then BA = I.
Proof. First note that the proposition is true for complex matrices. Let A =
A1 + jA2 and B = B1 + jB2 , where A1 , A2 , B1 and B2 are n × n complex
matrices and j 2 = 1 . Then,
AB = (A1 B1 + A2 B2 ) + j (A1 B2 + A2 B1 ) = I.
From this we have
A2
A1
B1
B2
B2
B1
=
B1
B2
B2
B1
=
I
0
I
0
0
I
so
A1
A2
Since
get
A1
A2
A2
A1
A2
A1
and
B1
B2
B2
B1
B1
B2
B2
B1
A1
A2
.
are 2n × 2n complex matrices, we
A2
A1
=
I
0
0
I
.
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H.H. Kösal and M. Tosun
Adv. Appl. Clifford Algebras
Then we can write B1 A1 + B2 A2 = I, B1 A2 + B2 A1 = 0. So
(B1 A1 + B2 A2 ) + jB1 A2 + B2 A1 = I.
Thus we obtain BA = I.
Definition 3.3. Let A = A1 + jA2 ∈ Mn (H) where A1 and A2 are complex
matrices. We define the 2n × 2n complex matrices η (A) as
A1
A2
A2
A1
.
This matrix η (A) is called the complex adjoint (or adjoint) matrix of the
commutative quaternion matrix A.
Definition 3.4. Let A ∈ Mn (H) and η (A) be the adjoint matrix of A. We
define the q-determinant of A by detq (A) = det (η (A)) . Here det (η (A)) is
usual determinant of η (A) .
i j+k
1
k
the adjoint matrix of A is
⎛
Example. Let A =
be a commutative quaternion matrix. Then
i
0
⎜ 1
0
η (A) = ⎜
⎝ 0 1+i
0
i
⎞
0 1+i
0
i ⎟
⎟.
i
0 ⎠
1
0
detq (A) = det (η (A)) = −3 − 4i.
Note that η A(1) = η (A). But η A(2) = η (A) and η A(3) = η (A) in
general, for A ∈ Mn (H) .
Theorem 3.5. Let A, B ∈ Mn (H) . Then the following properties hold:
I. η (In ) = I2n ;
II. η (A + B) = η (A) + η (B) ;
III. η (AB)= η (A) η (B) ;
−1
IV. η A−1 = (η (A)) if A−1 exists;
† †
†
†
V. η A 1 = (η (A)) , but η A†2 = (η (A)) , η A†3 = (η (A)) in gen†
eral, (η (A)) is the conjugate transpose of η (A) ; VI. detq (AB) = detq (A) detq (B) , consequently detq A−1 = (detq (A))−1 ,
if A−1 exists;
Proof. I, II and IV can be easily shown. We will prove III , V and VI.
III. Let A, B ∈ Mn (H) , where A = A1 + jA2 and B = B1 + jB2 for the
complex matrices A1 , A2 , B1 and B2 . Then the adjoint matrices of A and B
are
A 1 A2
B1 B2
η (A) =
, η (B) =
.
A 2 A1
B 2 B1
If we calculate AB, then we have
AB = (A1 B1 + A2 B2 ) + j (A1 B2 + A2 B2 ) .
Vol. 24 (2014)
Commutative Quaternion Matrices
775
So far the adjoint matrix of AB we can write
η (A) =
On the other hand,
η (A) η (B) =
A1
A2
A2
A1
A1 B1 + A2 B2
A1 B 2 + A 2 B 2
B1
B2
B2
B1
A1 B 2 + A 2 B 2
A1 B 1 + A 2 B 2
=
A1 B1 + A2 B2
A1 B2 + A2 B2
.
A1 B2 + A2 B2
A1 B1 + A2 B2
.
Thus we have η (AB) = η (A) η (B) .
T
T
V. Let A = A1 +jA2 ∈ Mn (H) . Since A†1 = A1 +j A2 , adjoint matrix
of A†1 is
T T † A1
A2
1
η A
=
.
T T
A2
A2
On the other hand
T T A1
A2
A1 A2
†
.
and (η (A)) =
η (A) =
T T
A2 A1
A2
A2
†
Thus we obtain η A†1 = (η (A)) .
Moreover
T T † AT1
A1
A2
−AT2
2
=
η A
in general
=
T T
−AT2
AT1
A2
A2
η A
†3
T
A1
=
T
− A2
T − A2
=
T
A1
T
A1
T
A2
T A2
in general.
T
A2
VI. From Theorem (3.5-III) we obtain η (AB) = η (A) η (B) . And so
detq (AB) = det (η (AB)) = det (η (A) η (B))
= det (η (A)) det (η (B)) = detq (A) detq (B) .
−1
Here, if we let B = A−1 then we can find easily detq A−1 = (detq (A)) .
Definition 3.6. Let A ∈ Mn (H) and λ ∈ H. If λ holds the equation Ax = λx
for some nonzero commutative quaternion column vector x, then λ is called
eigenvalue of A. The set of eigenvalues
σ (A) = {λ ∈ H : Ax = λx, for some x = 0}
is called spectrum of A.
Theorem 3.7. An n × n matrix A with commutative quaternion coefficients
has 2n complex eigenvalues.
Proof. Let A = A1 + jA2 ∈ Mn (H) and λ ∈ C be an eigenvalue of A.
Therefore there exist a nonzero column vectors, such that Ax = λx. This
implies
(A1 + jA2 ) (x1 + jx2 ) = λ (x1 + jx2 )
A1 x1 + A2 x2 = λx1 and A1 x2 + A2 x1 = λx2 .
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H.H. Kösal and M. Tosun
Adv. Appl. Clifford Algebras
Using these equations we way write
A2
A1
A1
A2
x1
x2
=λ
x1
x2
.
Therefore, commutative quaternion matrix A has 2n complex eigenvalues.
Corollary 3.8. For every A ∈ Mn (H) ,
σ (A) ∩ C = σ (η (A))
where σ (η (A)) = {λ ∈ C : η (A) y = λy, for some y = 0} , is the spectrum
of the complex adjoint matrix of A.
Theorem 3.9. Let A = A1 +jA2 ∈ Mn (H). Then the commutative quaternion
λ = λ1 + jλ2 is a eigenvalue of A if and only if there exist x1 , x2 ∈ C1n ,
(x1 = 0, x2 = 0) such that
A1 − λ 1 I n
A2 − λ 2 I n
A2 − λ 2 I n
A1 − λ 1 I n
x1
x2
=
0
0
.
Proof. Let A = A1 + jA2 , Then λ = λ1 + jλ2 is eigenvalue of A if and only
if there exists x1 , x2 ∈ C1n , (x1 = 0, x2 = 0) such that
(A1 + jA2 ) (x1 + jx2 ) = (λ1 + jλ2 ) (x1 + jx2 ) .
This may be written as
A1 x1 + jA1 x2 + jA2 x1 + A2 x2 = λ1 x1 + jλ1 x2 + jλ2 x1 + λ2 x2
(A1 x1 + A2 x2 ) + j (A1 x2 + A2 x1 ) = λ1 x1 + λ2 x2 + j (λ1 x2 + λ2 x1 )
which is equivalent to the following equations
(A1 − λ1 In ) x1 + (A2 − λ2 In ) x2 = 0
(A1 − λ1 In ) x2 + (A2 − λ2 In ) x1 = 0.
Using these obtained equations, we may write
A1 − λ 1 I n
A2 − λ 2 I n
A2 − λ 2 I n
A1 − λ 1 I n
x1
x2
=
0
0
.
Theorem 3.10 (Gershgorin theorem for commutative quaternion matrices).
Let A = (aij ) ∈ Mn (H) . Then
σ (A) ⊆
n
{q ∈ H : q − aii ≤ Ri }
i=1
where Ri =
n
j=1, i=j
aij .
Proof. Let λ be any eigenvalue of A = (aij ) ∈ Mn (H) and x = 0 be the
corresponding eigenvector then Ax = λx. Suppose xi is component of x such
that xi ≥ xj for all j then we have xi > 0. On the other hand, λxi
corresponds to the i th component of vector Ax which means that
n
λxi =
aij xj .
j=1
Vol. 24 (2014)
Commutative Quaternion Matrices
777
So, we may write
λxi − aii xi =
n
n
aij xj ⇒ (λ − aii ) xi =
j=1, i=j
aij xj .
j=1, i=j
Taking norm of both sides of the obtained equation gives
n
(λ − aii ) xi = aij xj j=1, i=j
⇒ (λ − aii ) xi ≤
⇒ (λ − aii ) xi ≤
⇒ (λ − aii ) ≤
Thus we have
σ (A) ⊆
n
n
n
j=1, i=j
n
j=1, i=j
j=1, i=j
aij xj aij xj aij = Ri .
{q ∈ H : q − aii ≤ Ri }.
i=1
i i+j
1
k
Then the adjoint matrix of A is
⎛
Example. Let A =
; A is a commutative quaternion matrix.
⎞
0 1+i
0
i ⎟
⎟.
i
0 ⎠
1
0
√
√
The eigenvalues of η (A) are σ (η (A)) = i ± 1 + i , ± −2 − i . The
Gerschgorin disks are
√
D1 = z ∈ C : z − i ≤ 2 , D2 = {z ∈ C : z ≤ 2} .
i
0
⎜ 1
0
η (A) = ⎜
⎝ 0 1+i
0
i
Thus, we obtain σ (A) ∩ C ⊆ D1 ∪ D2 .
Theorem 3.11. Let A ∈ Mn (H) . The following are equivalent:
I. A is invertible,
II. Ax = 0 has a unique solution,
III. det (η (A)) = 0 i.e η (A) is an invertible,
IV. A has no zero eigenvalue.
Proof. (I.) ⇒ (II.) This is trivial.
(II.) ⇒ (III.) Let A = A1 + jA2 , x = x1 + jx2 where A1 , A2 are complex
matrices and x1 , x2 are complex column vectors. Then
Ax = (A1 x1 + A2 x2 ) + j (A1 x2 + A2 x1 ) .
From Ax = 0 we can write
(A1 x1 + A2 x2 ) = 0
and
(A1 x2 + A2 x1 ) = 0.
778
H.H. Kösal and M. Tosun
Adv. Appl. Clifford Algebras
So we get that
Ax = 0 if and only if
A1
A2
A2
A1
x1
x2
= 0.
T
= 0. Since Ax = 0 has a unique solution,
That is, η (A) x1 x2
T
η (A) x1 x2
= 0 has a unique solution. Thus, since η (A) is complex
matrix, η (A) is invertible.
(II.) ⇒ (IV.) This is trivial.
(III.) ⇒ (I.) If η (A) is invertible, then for A = A1 +jA2 there exist a complex
B1 B 2
matrix
such that
B2 B1
B1
B2
B2
B1
A1
A2
A2
A1
=
I
0
0
I
.
Here we obtain B1 A1 +B2 A2 = I and B1 A2 +B2 A1 = 0. Using this equation
we can write
(B1 A1 + B2 A2 ) + j (B1 A2 + B2 A1 ) = I.
That is BA = I for B = B1 +jB2 . So A is invertible commutative quaternion
matrix by Theorem (3.2).
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Commutative Quaternion Matrices
779
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Hidayet Hüda Kösal and Murat Tosun
Sakarya University
Faculty of Arts and Sciences
Department of Mathematics
Sakarya
Turkey
e-mail: hhkosal@sakarya.edu.tr
tosun@sakarya.edu.tr
Received: July 26, 2013.
Accepted: November 11, 2013.