On the Fundamental Spinor Matrices of Real Quaternions

TÜLAY ERİŞİR1, EMRAH YILDIRIM2

1Department of Mathematics, Faculty of Arts and Sciences,

Erzincan Binali Yıldırım University,

24002, Erzincan,

TURKEY

2Department of Mathematics, Graduate School of Natural and Applied Sciences,

Erzincan Binali Yıldırım University,

24002, Erzincan,

TURKEY

Abstract: - In this study, the real quaternions and spinors are studied. The motivation of this study is to express

the Hamilton matrices of real quaternions more shortly and elegantly, namely spinors. Therefore, firstly, two

transformations between real quaternions and spinors are defined. These transformations are defined for two

different spinor matrices corresponding to the left and right Hamilton matrices since the quaternion product is

not commutative. Thus, the fundamental spinor matrix corresponding to the fundamental matrix of real

quaternions is obtained and some properties are given for these spinor matrices. Finally, the eigenvalues and

eigenvectors of the fundamental spinor matrix are obtained.

Key-Words: - Spinors, Quaternions, Hamilton matrices, Spinor metrices

Received: July 11, 2023. Revised: October 8, 2023. Accepted: October 25, 2023. Published: November 15, 2023.

1 Introduction

Quaternions were obtained by generalizing complex

numbers to develop a new number system. The

studies, [1], [2], [3], introduced a new multiplication

operation with this into vector algebra and obtained

the quaternions that may be possible in its division.

Moreover, real quaternions have no commutative

property. Quaternion product is a combination of the

scalar multiplication, the Euclidean inner product,

and the vector product. Thus, the set of real

quaternions is a two-dimensional vector space on

the set of complex numbers and a four-dimensional

vector space on the set of real numbers. Today, the

matrices are used in many branches of science and

have a great importance. For this reason, the

relationship of matrices with quaternions has

attracted much attention. Since the quaternion

product is not commutative, the product of the right

and the product of the left with another quaternion

are not equal. This is paired matrices with

44x

the

type quaternions with two separate multiplications,

thanks to a transformation. Thus, the left product

represents the left matrix representation of a

quaternion, and the right product represents the right

matrix representation of a quaternion. These matrix

representations are called Hamilton matrices

corresponding to the Hamilton operators of the real

quaternions. On the other hand, a lot of studies have

been done about matrices whose elements are

composed of quaternions. The studies on

quaternions date back to 1936. The study, [4], gave

the concept of similarity for matrices whose

elements are real quaternions. Later, the study, [5],

made a study on the eigenvalue and diagonalization

of quaternion matrices. In addition, the study, [6],

showed that every quaternion matrix, including a

square, has a characteristic root, and in addition,

similar matrices have the same characteristic root.

The study, [7], gave one of the most important

studies on quaternion matrices.

With the introduction of Hamilton in 1843,

quaternions have found many uses until today.

Quaternions, which provide great convenience in

engineering fields apart from geometry and algebra,

are also of great importance in the mathematics of

today's technology. It is also used in computer

graphics, physics, mechanics, kinematics, computer

games, animations, and digital imaging. Quaternions

are of great importance in geometry, especially in

the representation of the rotation of objects in 3-

dimensional space. Quaternions also have many

uses in physics. The use of complex numbers in

mechanical and electrical applications, especially in

circuit analysis, limits the applications since they are

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two-dimensional. In 3-dimensional applications, in

cases where vectors are insufficient in some

applications, quaternions add a fourth dimension to

applications, providing great convenience. The most

important of the fields where quaternions find

applications in physics is Einstein's special and

general relativity theories. Quaternions are used to

describe electron spin in quantum mechanics.

Quantum operators operating on a spinor can be

represented by quaternions considering the

relationship of

22x

matrices with quaternions. With

the help of this approach, B. L. Van der Waerdan

developed the mathematical formula of spinors in

1930. Pauli in 1927 and Dirac in 1938 demonstrated

spinor equations to describe the electron spin

physically.

The introduction of spinors is one of the most

difficult topics in quantum mechanics. Even if spin-

1/2 is considered, some fundamental aspects of

spinors, such as the effects of rotation on spinors,

turn out to be difficult to explain. According to

physicists, spinors are multilinear transformations.

Thanks to this feature, spinors are mathematical

entities somewhat like tensors and allow a more

general treatment of the notion of invariance under

rotation and Lorentz boosts. For mathematicians,

spinors are vectorial objects and their multilinear

features do not play any role. In addition, spinors

have one index. In discussing vectors and tensors,

there are two ways, in which we can proceed: the

geometrical and analytical. To use the geometrical

approach, we describe each kind of quantity in

terms of its magnitudes and directions. In the

analytical treatment, we use components. The study,

[8], while investigating linear representations of

simple groups developed the most general

mathematical form of spinors. In geometric terms,

the study, [9], introduced spinors. The study, [9],

developed the spinor theory geometrically by giving

only the geometric definition of spinors. Then, the

spinor formulation of curves is given by the study,

[10], considering Frenet vectors of curves in three-

dimensional Euclidean space. That study is an

important study for the relationship between curve

theory and spinors in differential geometry. Pauli

matrices, which are a basis

 

2SU

, and spinor

algebra, which have two complex components,

provide a nice representation of rotations in three-

dimensional real space. In this context, the study,

[11], established a new relationship between

quaternions and spinors and expressed quaternion

kinematics with spinors. In the study, [11], a one-to-

one and linear relationship was established between

spinors and real quaternions, and spinor formulation

and thus spinor kinematics of spins represented by

real quaternions were obtained. On the other hand,

the study, [12], investigated quaternions and spinors

in quantum mechanics by establishing a relationship

between

 

3SO

and

 

2SU

. The study, [13],

obtained a main expression of quaternions with

matrices by

22x

by considering the isomorphism

between real quaternions and spinors. Moreover, the

properties of the fundamental real matrix associated

with a quaternion were investigated and a frequently

considered quaternion equation was examined, from

which the nth power of a quaternion can be

determined, [14]. Therefore, the spinor

representations in Euclidean 3-space were studied

using different frames such as Darboux, Bishop, q-

frame, [15], [16], [17]. The study, [18], obtained the

Frenet spinor equations of Lie groups in Euclidean

space with a bi-invariant metric and gave some

special situations for these Lie groups with three-

dimensional. Later, studies on spinors in this field of

differential geometry focused on special curves. The

study, [19], revealed the spinor representations of

the involute-evolute curves and the relationship

between these spinors. Then, Bertrand curves were

represented by spinors in the complex plane, and the

study, [20], proved the relationship between spinors

corresponding to these Bertrand curves. After that,

the successor curve couple corresponded to two

different spinors, and geometric interpretations were

derived, [21]. In addition to that, the spinor

representations of some curve pairs selected in

Minkowski space were obtained, [22], [23], [24],

[25]

The motivation of this study is to obtain a new

and easier matrix representation of the Hamilton

matrices corresponding to the Hamilton operators of

the real quaternions. For this, first, considering the

isomorphism between spinors and quaternions, the

spinor matrices corresponding to the right and left

Hamilton matrices of real quaternions have been

created. Since the quaternion product is not

commutative, two separate spinor matrices

corresponding to these Hamilton matrices have been

formed. These matrices have been called the left and

right Hamilton spinor matrices. Moreover, some

properties of these right and left Hamilton spinor

matrices have been given. Consequently,

considering the left Hamilton spinor matrix as the

fundamental spinor matrix some theorems and

results about the eigenvalues and eigenvectors of the

fundamental spinor matrix have been obtained.

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2 Preliminaries

2.1 Quaternions and Spinors

In this section, we give some propositions and

theorems about the real quaternions and spinors. Let

0 1 2 3

q q q q q    i j k

be real quaternions in the set

of real quaternions

H

where

0 1 2 3

, , ,q q q q 

and

3

,,i j k

such that

1= = = ii jj kk

,

=ij ji k

,

==jk kj i

,

==ki ik j

. In this

case, the set of real quaternions

H

can be

expressed

3

0 1 2 3 0 1 2 3

{ , , , , , , , }.q q q q q q q q q q       i j k i j kH R R

Now, we assume that the real quaternion

0 1 2 3

q q q q q    i j k

is

qq

qSV

where the

scalar part of

q

is

0q

Sq

and the vector part of

q

is

1 2 3

qq q q  V i j k

, [26]. Let any two real

quaternions be

pp

pSV

and

qq

qS  VH

.

Therefore, the addition of these real quaternions is

expressed

       

p p q q p q p q

p q S S S S        V V V V

where

:  H H H

. Moreover, the scalar

product of the real quaternion

qq

qS  VH

is

written by

 

q q q q

q S S

   

   VV

where



R

and

:R H H

. Consequently, with the

aid of these operations, we say that the system

 

, , , ,.,HR

is a real vector space, briefly, this

space can be denoted by

H

where

 

p

S , , ,1 i j kH

and

 

dim 4H

, [26]. Assume

that any two real quaternions are

,

p p q q

p S q S    VVH

. In this case, this

quaternion product is

,

p q p q p q q p p q

pq S S S S     V V V V V V

where

“

,

” and “



” are Euclidean scalar and vector

products in

3

E

. We know that the product of two

quaternions is a quaternion, the product of

quaternions has the properties of associative and

distributive. But the quaternion product is not

commutative. Thanks to these properties, the system

{ , , , ,., }HR

is an associative algebra, [26]. Let

any real quaternion be

qq

qS  VH

. Therefore,

the conjugate of the quaternion

q

is expressed as

*qq

qSV

. On the other hand, the norm of the

real quaternion

q

is defined by

* 2222

0 1 2 3

()N q qq q q q q    

. In this case, if

the norm of the quaternion

qH

is

( ) 1Nq

then,

this quaternion is called the unit quaternion. In

addition that, the inverse of the real quaternion

q

is

*

1

2()

q

qNq



where

0q

, [26].

On the other hand, assume that the real quaternion

qH

. Therefore, we can write this quaternion as

   

0 1 2 3 0 1 2 3

q q q q q q q q q       i j k i i j

and we say that

 

1 2 1 0 1 2 2 3

,q z z z q q z q q       j i iHC

is

isomorphic with

2

C

since the real quaternion

q

matches with the complex number

2

12

zzjC

,

[26], [27]. In addition to that, the real quaternion can

be written as

   

0 1 2 3 0 1 2 3 1 2 .q q q q q q q q q z z         i j k i j ji

In this case,

H

has the basis

 

,1j

. Therefore, the

transformation

()

q

T p pq

is linear and the

2×2

matrix corresponding to this linear transformation is

0 1 2 3

2 3 0 1

q q q q







  



ii

,

[26]. The algebra of the real quaternions

H

contains infinite sub-algebras derived from bases

such as

     

, , ,......, , ,1 i 1 j 1 k

, [28]. Therefore,

the set of quaternions

H

can be written according

to many bases. Moreover, the study, [28], wrote the

real quaternion

q

as

   

0 1 2 3 0 2 1 3

q q q q q q q q q       i j k j i j

with the aid of the basis

 

,1i

and obtained the

complex matrix

0 1 1 3

1 3 0 2

q q q q

  









jj

.

Assume that

33

C

is the complex vector space and

the vector

3

1 2 3

( , , )x x xx3

C

is isotropic vector

where

222

1 2 3 0xxx  

. The set of the isotopic

vectors in the complex vector space

33

C

corresponds to a surface with two dimensional in

23

C

. If this surface with two dimensional is

parameterized by the complex numbers

1



and

2



then, the equations

,

22

1 1 2

x = -



22

2 1 2

x = ( + )



i

,

222

3 1 2

x =-



are provided where

i

is the complex

unit,

21i

and

12

,



C

. It can be easily seen

that every isotopic vector in the complex vector

space

33

C

corresponds to two vectors in

2

C

such

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that

12

( , )



and

12

( , )





. On the contrary, these

both vectors are given in this way in

2

C

correspond

to a single isotropic vector

x

. Cartan expressed that

the complex vector with two dimensional such that

1

2











is called as spinor, [9]. Moreover, the

study, [9], gave that the conjugate of the spinor

12

( , )

  



is

C



i

.

 

SO 3

, the group of

rotations about the origin in a three-dimensional real

vector space

3

R

, is homomorphic to

 

2SU

, the

group of unitary matrices with two dimensional

22

. The elements of group

 

SO 3

move vectors

in vector space

3

R

, while elements of group

 

2SU

move vectors with two complex

components, i.e. spinors, [10], [29], [30]. With the

aid of this homomorphism, the study, [31], paired an

isotropic vector with a spinor

1

2











. Now, we

assume that



is a vector whose Cartesian

components are the complex symmetric

22

matrices such that

1 2 3

1 0 0 0 1

,,

0 1 0 1 0

  



     

  

     



     

i

.

These matrices are produced in Pauli matrices

1 2 3

0 1 0 1 0

,,

1 0 0 0 1

P P P



     

  

     



     

i

with the

help of the matrix

01

10

C







, [31]. Therefore, the

isotropic vector

3

+abCi

can be written by any

spinor



such that

=t

+

 

abi

. In addition to that,

the real vector

3

cR

, orthogonal with the vectors

3

,ab R

, can be expressed

 



c

in terms of

the spinor



where

""



is the mate of the spinor

such that

C





, [31].

The study, [11], gave the relation between real

quaternions and spinors with the transformation

:fHS

such that

30

0 1 2 3

12

( ) ( ) qq

q f q f q q q q qq







       





i j k i

i

.

In addition, the study, [11], obtained that the

bijective transformation

f

provides the following

equations

i) ( ,

ii) ( ) ( ),

f q+ p)= f(q)+ f(p)

f q f q

  

R

for

,pqH

. Therefore, the transformation

f

is

linear.

2.2 The Hamilton Operators of Real

Quaternions

We assume that any real quaternion is

0 1 2 3

q q q q q     i j k H

and the linear

transformation

h

is as

 

:h

p h p qp





  

HH

. (1)

In this case, we can write the matrix corresponding

to basis





, , , 1 i j k

0 1 2 3

1 0 3 2

2 3 0 1

3 2 1 0

()

q q q q

Hq q q q q

q q q q



  



















with the aid of the transformation

h

. Similarly,

with the aid of the linear transformation

 

( ):

.

hq

p h p pq





  

HH

(2)

Therefore, the matrix corresponding to the basis





, , , 1 i j k

can be obtained as

0 1 2 3

1 0 3 2

2 3 0 1

3 2 1 0

()

q q q q

Hq q q q q

q q q q



  



















according to the transformation

h

. Here, the

matrices

()Hq



and

()Hq



are called the left and

right Hamilton matrices corresponding to the

Hamilton operators

h

and

h

of the real

quaternion

q

, [27], [32]. On the other hand, we

suppose two real quaternion

0 1 2 3

p p p p p    i j k

and

0 1 2 3

q q q q q    i j k

. In this case, the matrix

form of the product

qp

of these real quaternions is

0 0 1 1 2 2 3 3

1 0 0 1 3 2 2 3

2 0 3 1 0 2 1 3

3 0 2 1 1 2 0 3

0 1 2 3 0

1 0 3 2 1

2 3 0 1 2

3 2 1 0 3

()

q p q p q p q p

qp q p q p q p q p

q p q p q p q p

q q q q p

q q q q p H q P

q q q q p



  





  





  



  



  

   

   



   



   



   



   

.

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Similarly, the matrix form of the product

pq

of

these real quaternions is

0 0 1 1 2 2 3 3

1 0 0 1 3 2 2 3

2 0 3 1 0 2 1 3

3 0 2 1 1 2 0 3

0 1 2 3 0

1 0 3 2 1

2 3 0 1 2

3 2 1 0 3

()

q p q p q p q p

pq q p q p q p q p

q p q p q p q p

q q q q p

q q q q p H q P

q q q q p



  





  





  



  



  

   

   



   



   



   



   

, [27].

Moreover, the representation

()Hq



of a quaternion

q

is well-known in algebra. Since

()Hq



plays a

crucial role in our subsequent considerations, we

call it as the fundamental matrix.

Theorem 2.1: Let any real quaternion be

qH

. In

this case, the Hamilton matrices

H

and

H

produced from the operators

h

and

h

are

orthogonal, [27].

Theorem 2.2: The product of Hamilton matrices

H

and

H

are commutative. Therefore, there is

equality

( ) ( ) ( ) ( )H q H p H p H q

   



, [27].

Theorem 2.3: Let any two real quaternions be

,pqH

and



R

. Hence, the Hamilton matrices

H

and

H

provide the following properties;

2()

i) p= q H (p)= H (q) H (p)= H (q)

ii) H (p+q)= H (p)+ H (q),

H (p+q)= H (p)+ H (q)

iii) H (pq)= H (p)H (q), H (pq)= H (q)H (p)

iv) H ( q )= H (q), H ( p)= H (q)

2

v) det H (q) = N (q), det H (q) = N q

vi)

   

   



  

  

     

   



   

  

   

   

 

1

1, ( ) 0

H (q )= H (q) ,

H (q )= H (q) N q



   



   

3 Main Theorems and Proofs

In this section, we establish a relationship between

quaternions and spinors and we give spinors

corresponding to real quaternions. Then, we express

the spinor matrices corresponding to Hamilton

matrices of real quaternions. In addition to that, we

calculate the eigenvalues and eigenvectors of these

spinor matrices, which we call Hamilton spinor

matrices, after giving some properties of these

matrices. Consequently, we obtain some

conclusions.

3.1 Spinor Representation of Real

Quaternions

Let

0 1 2 3

q q q q q    i j k H

be an arbitrary real

quaternion where

0 1 2 3

, , ,q q q q 

,

3

,,i j k

, and

H

is real quaternion space. In this case, we can

write the real quaternion

q

with regard to the basis

 

k,i

with left multiplication as

3 0 1 2

( ) ( )q q q q q   iiki

(3)

where

==iji k

and since

21i

we can consider

that

i

is imaginer unit. Therefore, the quaternion

q

can be expressed in terms of two complex numbers

1 3 0 2 1 2

q q q q



     iiC

. So, we can give the

following definition.

Definition 3.1: Let

0 1 2 3

q q q q q   i j k

be a real

quaternion in

H

. In this case, the set of real

quaternions is defined by

 

1 2 1 3 0 2 1 2

,q q q q q

   

       kiHCii

(4)

where ‘’ ‘’ is a complex conjugate.

As a result of Definition 3.1, according to the

expression given in equation (4) the following

transformation can be given.

Let the real quaternion be

q

written by in terms of

the spinor



. Therefore, we can give the

transformation between quaternions and spinors as

30

0 1 2 3

12

:

( ) ( )

f

qq

q f q f q q q q qq









       





i j k

HS

i

(5)

with aid of the equation (3), [11]. As is known, the

vector space

H

is isomorphic to the space

2

C

,

[33]. Therefore, the transformation

f

defined in

equation (5) is isomorphism.

Now, we give the following definition about the

conjugate of spinors based on the relationship

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between quaternions and spinors in the equation the

transformation

f

.

Definition 3.2: Suppose that the quaternion

0 1 2 3

q q q q q    i j k H

matches the spinor

30

12

qq















S

i

. In this case, the four different

definitions of conjugate for spinors;

i) The complex conjugate



of the spinor

1

2











S

corresponding to the real quaternion

qH

is defined by

1 3 0

12

2

.

qq





 















i

ii) The spinor

*



corresponding to the quaternionic

conjugate

*

0 1 2 3

q q q q q   i j k

of the real

quaternion

qH

is defined by

30

*1

12

2

.

qq





 





  









i

iii) The spinor conjugate



of the spinor



S

by

given, [9], is defined by

21

12

03

21

01 .

10

qq





    



  

    

 



 

   

   

i

ii

i

(6)

iv) The mate of the spinor



S

by given, [31], is

defined by

12

30

21

01

10

qq





    





   

    

 



 

   

   

i

.

Therefore, the following corollaries can be given.

Corollary 3.1: Let

q

be an arbitrary real

quaternion and



be the spinor corresponding to

this quaternion

q

. There is the relationship



i

between the spinor conjugate and the mate of the

spinor



.

Corollary 3.2: Consider that the spinor



S

corresponds to the real quaternion

qH

. In this

case, the spinor equation of the norm of

q

is given

that

 

t t t t

N C C

    



i

where

01

10

C







.

Proof: Suppose that the spinor



S

corresponds

to the real quaternion

qH

. Therefore, we write

1

21

2

2222

1 1 2 2 0 1 2 3

01

10

()

tt

C

qqqqN



    





 



 





 



      

and

 

2222

0 1 2 3 ()

t

tt t t t t

C C C C C

qqqqN

     



  

    

i i i

where

01

10

C







.

3.2 Hamilton-Spinor Matrices

The quaternion product is not commutative in the

quaternion algebra therefore, the Hamilton matrices

corresponding to the right and left quaternion

products are different. In this case, in this section,

we obtain the spinor matrices for the right and left

Hamilton matrices, separately. Moreover, we call

these spinor matrices the left and right Hamilton

spinor matrices. Then, we give some properties and

theorems for these spinor matrices.

We know that the set of quaternions can be

written as in equation (4). In this case, the operator

in equation (1) is linear transformation therefore,

this linear transformation corresponds to a matrix.

Therefore, the following theorem can be given.

Theorem 3.3: Let

0 1 2 3

q q q q q    i j k H

be

an arbitrary real quaternion written as

12

q



ki

where

1 3 0 2 1 2

,q q q q



     Cii

. Therefore,

the left Hamilton-spinor matrix corresponding to the

left Hamilton matrix of quaternion

qH

is given

by

0 3 2 1

2 1 0 3

.

L

q q q q



  









ii

(7)

Proof: Suppose that the real quaternion

0 1 2 3

q q q q q    i j k H

is written as

12

.q



ki

In this case, if we consider Hamilton

operator in equation (1) then, we obtain

     

1 2 1 2

h

   

   k k k i k k k i k

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(8)

and

 

   

1 2 2 1

h

   

    i k i i i k k i k

(9)

where

11



kk

,

22



kk

,

11



ii

and

22



ii

. Therefore, we find that the matrix form

of the equations (8) and (9) is

0 3 2 1

2 1 0 3

L

q q q q



  









ii

.

This matrix

L



is called the left Hamilton spinor

matrix corresponding the left Hamilton matrix of

real quaternion

qH

. The proof is completed.

Moreover, since

()Hq



is the fundamental matrix

for the real quaternion

q

we can call the left

Hamilton spinor matrix

L



the fundamental spinor

matrix.

Corollary 3.4: For the fundamental spinor matrix

L



the statement

() L

q p H q P





  

is provided where



is the spinor corresponds to

real quaternion

p

with the aid of the transformation

f

, [11].

Now, for the right Hamilton spinor matrix, we need

to define a new transformation similar to the

equation (5). For this, we can write that the real

quaternion

q

with regard to the basis

 

k,i

as

3 0 1 2

( ) ( )q q q q q   iiki

(10)

with right multiplication where

==iji k

.

Therefore, the quaternion

q

can be expressed in

terms of

1 3 0 2 1 2

,q q q q



    iiC

. In this

case, the set of real quaternions can be defined by

 

1 2 1 3 0 2 1 2

,q q q q q

   

       kiHCii

where ‘’ ‘’ is a complex conjugate. Moreover,

similar to the transformation

f

, with the aid of

equation (10) the transformation

*

f

can be given as

*

30

* 0 1 2 3 *

12

:

( ) ( ) .

f

qq

q f q f q q q q qq









       





i j k

HS

i

(11)

Theorem 3.5: Let the real quaternion

qH

be

1 2 1 2

q

   

   k i k i

where

1 3 0

qq



i

and

2 1 2

qq



  Ci

. The right Hamilton spinor matrix

of this quaternion is

0 3 2 1

2 1 0 3

.

R

q q q q









  



ii

(12)

Proof: Assume that the real quaternion

0 1 2 3

q q q q q    i j k H

is

12

q



ki

. Then,

with the aid of the Hamilton operator in equation (2)

we can write

 

1 2 1 2

((h

   

   k k k k i k )k k )i

and, similarly

 

1 2 2 1

((h

   

    i i k i i k )k k )i

where

11



kk

,

22



kk

,

11



ii

and

22



ii

. Consequently, we get

0 3 2 1

2 1 0 3

R

q q q q









  



ii

.

Therefore, the matrix

R



is called the right

Hamilton spinor matrix corresponding to the left

Hamilton matrix of the real quaternion

q

.

Corollary 3.6: For the right Hamilton spinor matrix

we can write

*

() R

p q H q P





  

where

*



is the spinor corresponding to the real

quaternion

p

with the aid of the function

*

f

.

Now, we give the relationship between the Hamilton

spinor matrices

R



,

L



and the Pauli matrices.

Theorem 3.7: Let

0 1 2 3

q q q q q    i j k H

be

any real quaternion and

R



and

L



be the left and

right Hamilton spinor matrices corresponding to the

real quaternion

q

. Therefore, these spinor matrices

can be written as

0 2 1 1 2 2 3 3Lq I q P q P q P



   i i i

and

0 2 1 1 2 2 3 3Rq I q P q P q P



   i i i

where

2

22

IC

is unit matrix with

22

and

2

1 2 3 2

,,P P P C

are Pauli matrices.

Proof: Let the left Hamilton spinor matrix be

L



.

Then, from the equation (7) we get

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1

2

1

2

3

10

1, 01

01

,10

0

,1

10

,.

01

L

i

L

j

L

k

L

for q I

for q i P

for q j P

for q k P





  







    









    







    







ii

i

ii

i

ii

Hence, we have

0 2 1 1 2 2 3 3Lq I q P q P q P



   i i i

.

Similarly, for the right Hamilton spinor matrix in

equation (12) we obtain for

1q

1

2RI





, for

qi

1

i

RP



i

, for

qj

2

j

RP



i

and for

qk

3

k

RP



i

. Consequently, the equation

0 2 1 1 2 2 3 3Rq I q P q P q P



   i i i

is found.

Theorem 3.8: Let any two real quaternions be

,pqH

and the left and right Hamilton spinor

matrices corresponding to these quaternions be

,

LR



and

,

LR



, respectively. Then, the

following statement is provided;

 

**

*

22

1 * 1

*

( ) , ( ) ,

) ( ) , ( ) ,

( ) , ( ) ,

1 , 1 ,

ˆ,,

( ) , ( ) .

L L L R R R

L L R R

LR

L L L L R R R R

i) = + = +

ii = =

iii) = =

iv) I I

v) = P = P

vi) = =

       

   

   

   

       







 ii

where

,



R C1

and

2

0





  





C

.

Proof: Assume that

,

LR



and

,

LR



, are the left

and right Hamilton spinor matrices corresponding to

any two real quaternions

p

and

q

, respectively.

Therefore, we can give the proof.

i) Let

,



S

be two spinors corresponding to

the real quaternions

p

and

q

, respectively. Then,

with the aid of the transformation

f

in the equation

(5) we write

 

3 3 0 0

1 1 2 2

p q p q



    





  



i

and the left Hamilton spinor matrix of the spinor





is obtained as

         

       

0 0 3 3 2 2 1 1

2 2 1 1 0 0 3 3

0 3 2 1 0 3 2 1

2 1 0 3 2 1 0 3

.

L

LL

p q p q p q p q

p q p q p q + p q

p p p p q q q q

p p p p q q q + q



      

     

     



  









   

   

   

ii

i i i i



Similarly, for the right Hamilton spinor matrix, the

proof is completed easily.

ii) We assume that



R

and the spinor

30

12

.

qq



 











i

Then, we get the left Hamilton

spinor matrix of the spinor



 

0 3 2 1

2 1 0 3

0 3 2 1

2 1 0 3

.

L

q q q q

   

    

 

  









  











ii

On the other hand, we suppose that the spinor

*



corresponds to the real quaternion

q

with the aid of

the transformation

*

f

in the equation (11). In this

case, the right Hamilton spinor matrix of the spinor

*



is

 

0 3 2 1

*

2 1 0 3

0 3 2 1

2 1 0 3

.

R

q q q q

   

    

 







  











  



ii

iii) Suppose that the spinor



corresponds to the

real quaternion

qH

with the aid of the

transformation

f

and

ab



  Ci

. Then, we get

   

3 0 0 3

1 2 2 1

aq bq aq bq



    



  



i

and have the left Hamilton spinor matrix of the

spinor



     

   

 

0 3 3 0 2 1 1 2

2 1 1 2 0 3 3 0

0 3 2 1

2 1 0 3

0.

0

L

aq bq aq bq aq bq aq bq

q q q q a b



      

      

   

  







   

   

   

ii

i i i

Consequently, we obtain

 

L

 



where

0









. Similarly, the proof for the right

Hamilton spinor matrix can be obtained.

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iv) We consider that the real quaternion

q

is

1q

.

In this case, the spinor corresponding to

1q

is

1

0









. Consequently, we find that the left

Hamilton spinor matrix related to the spinor



is

2

10

01

LI











. Similarly, we get the right

Hamilton spinor matrix for the real quaternion

1q

as

2

10 .

01

RI











v) Let the spinor corresponding to the real

quaternion

q

be



and the spinor conjugate of the

spinor



be

21

03

ˆqq

qq













i

in the equation (6). In

this case, the left Hamilton spinor matrix of the

spinor

ˆ



can be obtained

 

1 2 3 0

3 0 1 2

0 3 2 1

1

2 1 0 3

ˆ

01 .

10

L

q q q q

q q q q P

q q q q









  



  











ii

Similarly, we obtain that the right Hamilton spinor

matrix of the spinor

*



is

 

1 2 3 0

*

3 0 1 2

0 3 2 1

1

2 1 0 3

01 .

10

R

q q q q

q q q q P

q q q q









  











   



ii

vi) For the spinors



and



corresponding to the

real quaternions

,pqH

we can write

 

0 3 2 1 30

2 1 0 3 12

0 3 3 0 2 1 1 2 0 0 1 1 2 2 3 3

2 3 1 0 0 1 3 2 2 0 1 3 0 2 3 1

.

L

q q q q pp

q p q p q p q p q p q p q p q p



   

 

      

      













ii

i

ii i

i

Therefore, we get the left Hamilton spinor matrix

corresponding to the spinor

L



S

as

 

0 3 2 1 0 3 2 1

2 1 0 3 2 1 0 3

.

L L L

L

q q q q p p p p

   

     

   



   

   

   

i i i i

Similarly, we obtain

 

0 3 3 0 2 1 1 2 0 0 1 1 2 2 3 3

*

2 3 1 0 0 1 3 2 2 0 1 3 0 2 3 1

R

q p q p q p q p q p q p q p q p



      

       







i

and consequently

 

*.

R R R

R

   



Proposition 3.9: The left and right Hamilton spinor

matrices

L



and

R



related with the real quaternion

q

are normal.

Proof: Suppose that



is the left Hamilton spinor

matrix related to the real quaternion

.q

Then, we

obtain

0 3 2 1 0 3 2 1

2 1 0 3 2 1 0 3

2222

0 1 2 3

2222

0 1 2 3

2

0

( ) .

t

LL

q q q q q q q q

q q q q q q q + q

qqqq

N q I



    

   

   

   

   













i i i i

Similarly, if we calculate the equation

t

LL



then,

we have

2

()

t

LL N q I





. Consequently, we get

tt

L L L L

   



and we say that the left Hamilton

spinor matrix

L



is normal. Similar to the left

Hamilton spinor matrix we get

2

()

tt

R R R R

N q I

   



and we see easily that the

right Hamilton spinor matrix

R



is normal.

Consequently, the proof is completed.

Now, we calculate the determinant of the left and

right Hamilton spinor matrices for the real

quaternion

0q

. Hence, we get

2

det( ) det( ) ( )

LR

Nq





.

In this case, we can give the following corollaries.

Corollary 3.10: The left and right Hamilton spinor

matrices

L



and

R



related with the real quaternion

0q

are regular.

Corollary 3.11: The inverses of the left and right

Hamilton spinor matrices where

0qH

are

0 3 2 1

1

2

2 1 0 3

0 3 2 1

1

2

2 1 0 3

1,

()

1.

()

L

R

q q q q

Nq

q q q q

Nq















  









+

ii

Corollary 3.12: The inverses of the left and right

Hamilton spinor matrices can be written as

   

1 * 1 *

*

22

11

,

( ) ( )

LR

N q N q

   





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respectively, where

 

*

L



and

 

*

*R



are the left

and right Hamilton spinor matrices related with the

quaternion conjugate

*

q

.

Especially, if we consider that the real quaternion

q

is unit then, we get

   

1 * 1 *

*

,

LR

   





Corollary 3.13: The left and right Hamilton spinor

matrices are unitary matrices. It should be

emphasized that

, (2)

LRSU





3.3 The Eigenvalues and Eigenvectors of the

Fundamental Spinor Matrix

In this section, we obtain the eigenvalue and

eigenvector of the fundamental spinor matrix (i.e.

left Hamilton spinor matrix)

L



. Similar equations

can be obtained for the right Hamilton spinor matrix

R



.

We consider that the real quaternion

q

is pure real

quaternion

qq

. Therefore, the fundamental spinor

matrix

p

L



corresponding to the pure quaternion

q

is obtained as

3 2 1

2 1 3

q q q

  









L

ii



.

Lemma 3.14: Let the fundamental spinor matrices

be

L



,

L



corresponding to the real quaternion

q

and the pure real quaternion

q

. Hence, the

relationship between these fundamental spinor

matrices is

02 .

LqI





L



Proof: Suppose that the fundamental spinor

matrices

L



and

L



are related with

q

and the pure

quaternion

q

, respectively. Then, we obtain that

0 3 2 1

2 1 0 3

3 2 1

0 0 2

2 1 3

10 .

01

L

q q q q

q q q

q q I

q q q



  









  



   



 



L

ii



Theorem 3.15: The eigenvalues of the fundamental

spinor matrix

L



are

10 ()qN



 qi

and

20 ()qN



 qi

where

0

qqq

and

q

is vectorial

part of the real quaternion

.q

Proof: We suppose that the

L



is the fundamental

spinor matrix. We know that the eigenvalues of the

spinor matrix

L



are obtained with the equation

L

  



where



C

are the eigenvalues.

Therefore, if we calculate the equation

 

2

det 0

LI





we get

 

0 3 2 1

2

2 1 0 3

det 0

L

q q q q

Iq q q q



 

   



  



  



ii

and we obtain that the characteristic polynomial is

2 2222

0 2 1 0 3

20qqqqq



     

.

Moreover, if we solve this second-order equation

then we get

1 0 2 0

( ), ( ) .q N q N



    qqC.ii

(13)

Corollary 3.16: Let

0 1 2 3

q q q q q   i j k

be any

real quaternion

L



be the fundamental spinor matrix

corresponding to the real quaternion

q

and

L



be

the fundamental spinor matrix corresponding to the

pure real quaternion

1 2 3

q q q  q i j k

. Therefore,

the eigenvalues of the fundamental spinor matrix

L



are found by adding

0

q

to the eigenvalues of

L



.

Proof: Let

L



and

L



be the fundamental spinor

matrices corresponding to the real quaternion

q

and

the pure real quaternion

q

respectively. Now, we

calculate the roots of the characteristic polynomial

 

2

det 0

LI







. Therefore, we get

 

2222

2 3 2 1

det 0

LI q q q



     



and

12

( ), ( ) .NN



   qqCii

(14)

Consequently, if we use equations (13) and (14) we

say that the eigenvalues of the fundamental spinor

matrix

L



are found by adding

0

q

to the

eigenvalues of the fundamental spinor matrix

L



.

Theorem 3.17: Assume that

L



is the fundamental

spinor matrix corresponding to the pure real

quaternion

q

. Moreover, we consider that the

eigenvalue of

L



is



. Therefore, the eigenvalue of

the spinor matrix

2

L



is

2



.

Proof: Suppose that the fundamental spinor matrix

corresponding to the real pure quaternion

q

is

L



.

Now, we find the spinor matrix

2

L



therefore, we

obtain

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 

222

2

3 2 1

2

222

3 2 1

0.

0

qqq NI

qqq



  

  



  



2

Lq



If we assume that the roots of the characteristics

polynomial

2

det( ) 0

LI







are

1



and

2



then,

we have

2

1,2 ( ).N



 q

(15)

Consequently, considering the equations (14) and

(15) we say easily that if the eigenvalue of the

spinor matrix

L



is



then, the eigenvalue of the

spinor matrix

2

L



is

2



.

Corollary 3. 18: The fundamental spinor matrix

L



corresponding to the real quaternion

q

is a non-

defective matrix.

Proof: We see that the number of eigenvalues of the

fundamental spinor matrix

L



is equal to its number

of dimensions. In this case, the fundamental spinor

matrix is a non-defective matrix.

Theorem 3.19: The eigenspaces corresponding to

the

 

10

qN



 qi

and

 

20

qN



 qi

eigenvalues of the fundamental spinor

L



are,

respectively,

 

L

  

S

and

 

L

  

S

where the spinors are

3

12

()qN

qq













q

i

,

3

12

()qN

qq













q

i

and

,

LL



are the fundamental

spinor matrices of the spinors

,.



Proof: Let the fundamental spinor matrix of the

spinor



S

corresponding to the real quaternion

qH

be

L



. Firstly, we obtain the eigenspace for

the eigenvalue

 

10 .qN



 qi

Then, we consider

the spinor

 

3

12

() .

qN

qq













qii

i

In this case, the

fundamental spinor matrix of the spinor



is

 

3 2 1

2 1 3

() .

()

L

N q q q

q q N q



    







q

ii

Moreover, let



S

be any spinor such that

30

12















i

. In this case, the spinor

L



S

is

calculated as

 

3 0 2 1 1 2 0 1 1 2 2 3 3 3

2 3 1 0 3 2 2 2 0 1 3 3 1 1

( ) ( )

L

q q q N q q q N

       



       

       



      







qq

i

On the other hand, for the fundamental spinor

matrix

L



we know that

 

3 2 1

12

2 1 3

() .

()

L

N q q q

Iq q N q



    





  



q

ii

If we calculate the equation

 

12LL

I

   



then,

we obtain

 

12

00.

0

LL

I

   



  





Consequently,

the eigenspace for the eigenvalue

 

10

qN



 qi

consists of the spinors

.

L



Similarly, the eigenspace for the eigenvalue

 

20

qN



 qi

of the fundamental spinor matrix

L



can be obtained. For this, we consider the spinor

 

3

12

() .

qN

qq













qii

i

Therefore, the fundamental

spinor matrix of the spinor



is

 

3 2 1

2 1 3

() .

()

L

N q q q

q q N q



    



  



q

ii

Consequently, if we make the necessary

arrangements we obtain

 

12 0

LL

I

   



and see

that the eigenspace for the eigenvalue

 

20

qN



 qi

consists of the spinors

.

L



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Volume 22, 2023

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WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.93

Tülay Eri

şi

r, Emrah Yildirim

E-ISSN: 2224-2880

865

Volume 22, 2023

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

All authors contributed to the study's conception and

design. Material preparation, data collection, and

analysis were performed by Tülay Erişir, and Emrah

Yildirim.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.93

Tülay Eri

şi

r, Emrah Yildirim

E-ISSN: 2224-2880

866

Volume 22, 2023