A Unified Algebraic Technique for Eigenvalues and Eigenvectors in Quaternionic and Split Quaternionic Mechanics

ISSN Online: 2327-4379 ISSN Print: 2327-4352

DOI: 10.4236/jamp.2019.79133 Aug. 29, 2019 1929 Journal of Applied Mathematics and Physics

A Unified Algebraic Technique for Eigenvalues

and Eigenvectors in Quaternionic and Split

Quaternionic Mechanics

Zhenwei Guo

1

, Gang Wang

1

, Dong Zhang

2

, Tongsong Jiang

1,2,3*

1_{School of Mathematical Science, Liaocheng University, Liaocheng, China}

2_{College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, China} 3_{School of Mathematics and Statistics, Heze University, Heze, China}

Abstract

This paper aims to present, in a unified manner, the algebraic techniques of eigen-problem which are valid on both the quaternions and split quaternions. This paper studies eigenvalues and eigenvectors of the v-quaternion matrices by means of the complex representation of the v-quaternion matrices, and derives an algebraic technique to find the eigenvalues and eigenvectors of v-quaternion matrices. This paper also gives a unification of algebraic tech-niques for eigenvalues and eigenvectors in quaternionic and split quaternio-nic mechaquaternio-nics.

Keywords

v-Quaternion, Complex Representation, Quaternion, Split Quaternion, Eigenvalues, Eigenvectors

1. Introduction

A quaternion, which was found in 1840 by William Rowan Hamilton [1], is in the form of q q q= 1+ 2i+q3j+q4k , i2 =j2 =k2 = −1, ijk= −1, in which

1, , ,2 3 4

q q q q ∈R, and ij= − =ji k_, jk= − =kj i_, ki= − =ik j_{. A split}

quater-nion, which was found in 1849 by James Cockle [2], is in the form of

1 2i 3j 4k

q q q= + +q +q _{, i}2_{= −1 , j}2_=k2_{=1 , ijk 1= , in which}

1, , ,2 3 4

q q q q ∈R, and ij= − =ji k_, jk= − = −kj i_, ki= − =ik j_{. Denote the sets}

of quaternions and split quaternions by H and Hs, respectively. They are all associative and non-commutative 4-dimensional Clifford algebra, but the Hs is a more complex algebraic structure, because the Hs contains zero divisors,

How to cite this paper: Guo, Z.W., Wang, G., Zhang, D. and Jiang, T.S. (2019) A Unified Algebraic Technique for Eigenvalues and Eigenvectors in Quaternionic and Split Quaternionic Mechanics. Journal of Applied Mathematics and Physics, 7, 1929-1940.

https://doi.org/10.4236/jamp.2019.79133

Received: July 28, 2019 Accepted: August 26, 2019 Published: August 29, 2019

Copyright © 2019 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

DOI: 10.4236/jamp.2019.79133 1930 Journal of Applied Mathematics and Physics nilpotent elements and nontrivial idempotents.

Quaternion and split quaternion are playing an increasingly important role in many fields. Till now, quaternions are widely used in programming video games, controlling spacecrafts, computer graphics, control theory, signal processing, quan-tum physics [3] [4] [5]. And there are many applications of split quaternions [6] [7] [8] [9], split quaternion is one of the tools for studying modern quantum mechanics in the physical, and Lorentzian rotations can be represented by split quaternions. Eigenvalue and eigenvector problems of quaternion and split quater-nion matrices have many applications and are the basic tools of many scientific researches, therefore they have strong research significance. For quaternion, in [10], the author studied the problems of eigenvalues and eigenvectors of quater-nion matrices by means of complex representation and compaquater-nion vector. In [11], the author applied the Lefschetz Fixed Point Theorem to show that every square matrix over the quaternions has right eigenvalues, in addition, the author classified these eigenvalues and discussed some of their properties. For split qu-aternion, in [12], by means of complex representation of a split quaternion ma-trix, the authors studied the problems of right split quaternion eigenvalues and eigenvectors of a split quaternion matrix. In [13], the authors discussed the properties of complex eigenvalues of a split quaternion matrix, and gave an ex-tension of Gershgorin theorem.

Because there are amazing relationships between quaternions and split quater-nions [14] [15], the purpose of this paper is to unify and generalize them to the general case, then use a common method to find their eigenvalues and eigen-vectors. Therefore, a new algebraic structure is defined, it is v-quaternion.

A v-quaternion is in the form of

2 2 2

1 2i 3j 4k, i 1, j k ijk ,

q q q= + +q +q = − = = =v (1.1)

in which 0≠ ∈v R , q q q q1, , ,2 3 4∈R , and ij= − =ji k , jk= − = −kj vi , ik= − = −ki j_{. Similarly, denote the sets of v-quaternions by} H_v. Obviously, the set of all v-quaternion is also a non-commutative 4-dimensional Clifford alge-bra. Specially, when v= −1, the ring of the v-quaternion Hv is the ring of the quaternion H; when v=1, the ring of the v-quaternion Hv is the ring of the split quaternion Hs.

Before start this paper, first introduce the necessary symbols and pre-liminary knowledge. Let R be the real number field, C R R= ⊕ i the complex number field, and Hv = ⊕R Ri⊕R Rj⊕ k the v-quaternion ring, in

which _i2_{= −1, j}2 _{=v, k}2 _{=v, ij= − =ji k, jk= − = −kj vi, ik= − = −ki j.}

If q q q= 1+ 2i+q3j+q4k∈Hv, q q q= 1− 2i−q3j−q4k is the conjugate of q,

then 2 2 2 2

1 2 3 4

qq q= +q vq vq− − , the norm of q is defined to be

2 2 2 2

1 2 3 4

q = qq = q +q −vq −vq . For any matrix

( )

m n

st v

A₌ a _∈H × _,

( )

st

A= a ,

( )

T

ts

A = a , *

( )

ts

A = a denote the conjugate, the transpose, the conjugate trans-pose of the matrix A, respectively. A v-quaternion matrix n n

(

)

v v

A_∈H × a_{∈H is}

said to be nonsingular if there exists a v-quaternion matrix n n

(

)

v v

DOI: 10.4236/jamp.2019.79133 1931 Journal of Applied Mathematics and Physics such that AB BA I ab ba= = n

(

= =1

)

. Two n n× v-quaternion matrices A and

B are said to be similar if there exists a nonsingular v-quaternion matrix P such that _{P AP B}−1 _{= , this is written as A B~ . For A∈C}m m× _,

( )

A

f

λ

denotes the characteristic polynomial of A.

In this paper, we study the problem of right eigenvalues and associated right eigenvectors of the v-quaternion matrix. A v-quaternion λ is said to be a right eigenvalue provided that Aα αλ= for nonzero vector α , and α is said to be an eigenvector related to the right eigenvalue λ.

2. Equivalence Classes of v-Quaternions

Two v-quaternions p and q are said to be similar if there exists a nonsingular v-quaternion x such that _{x px q}−1 _{= , written as}

~

p q. It is easy to find that p

and q are similar if and only if a unit v-quaternion u such that _{u pu q}−1 _{= , then}

two similar v-quaternions have the same norm. Obviously, ~ is an equivalence relation on the v-quaternions. Denote the equivalence classes containing the v-quaternion q by

[ ]

q .

It is easy to prove the following result by direct calculation. Proposition 2.1 If q q q= 1+ 2i+q3j+q4k is a v-quaternion, then

1) If

(

2 2

)

2

3 4 2 0

v q +q −q < _{, then construct a nonsingular v-quaternion x such} that

(

)

1 2 2 2

1 2 3 4 i,

x qx q− _{= +} q ₋v q ₊q _(2.1)

in which

(

(

)

)

(

)

2 2 2

1 3 2 3 4 2 4 3

2 2 2

2 3 4 ₂

4

i j

k.

x v q q v q q q v q q

q v q q _q

q

v v

= + − + + + +

 _{− +} 

 

+_ − + _

 

 

   

 

2) If

(

2 2

)

2

3 4 2 0

v q +q −q > , there must be v>0 at this point, then construct a nonsingular v-quaternion x such that

1 2 2 2

1 3 4 1 j,2

x qx q q q q

v

− _{= + + − (2.2)}

in which

(

2 2

)

2

(

)

3 3 4 2 2 4

2 2 2

3 3 4 2 2 4

i

1 _j 1 _k.

x vq v q q q q vq

q q q q q q

v v

= + + − − −

   

−_ + + − _ + − 

 

 

3) If 2

(

2 2

)

2 3 4

q =v q +q _{, when} q₂=0_{, then} q q= _{1, and}

1 1,

x qx q− _{= (2.3)}

in which x=1. When q2 ≠0, and this case also implies v>0, then construct

a nonsingular v-quaternion x such that

1

1 i 1 j,

x qx q

v

− _{= + + (2.4)}

in which x 1 vq3 vq4i 1

(

q2 1 k

)

v

DOI: 10.4236/jamp.2019.79133 1932 Journal of Applied Mathematics and Physics The following two special cases about quaternion and split quaternion comes from Proposition 2.1 with v= −1 and v=1, respectively.

Corollary 2.2 If q q q= 1+ 2i+q3j+q4k is a quaternion, then construct a

nonsingular quaternion x such that

1 2 2 2

1 2 3 4i,

x qx q− _{= +} q ₊q ₊q _(2.5)

in which

(

2 2 2

)

(

2 2 2

)

3 2 4 2 3 4 i 3j 2 4 2 3 4 k

x= − +q q q− + q +q +q +q + q +q − q +q +q .

Corollary 2.3 If q q q= 1+ 2i+q3j+q4k is a split quaternion, then

1) If 2 2 2

2 3 4

q >q +q , then construct a nonsingular split quaternion x such that

1 2 2 2

1 2 3 4i,

x qx q− _{= +} q ₋q ₋q

_(2.6)

in which

(

2 2 2

)

(

2 2 2

)

3 2 4 2 3 4 i 3j 2 4 2 3 4 k

x q= + q +q + q −q −q +q + q +q − q −q −q _.

2) If 2 2 2

2 3 4

q <q +q , then construct a nonsingular split quaternion x such that

1 2 2 2

1 3 4 2j,

x qx q− _{= +} q ₊q ₋q

_(2.7)

in which 2 2 2

(

)

(

2 2 2

)

(

)

3 3 4 2 2 4 i 3 3 4 2 j 2 4 k

x q= + q +q −q − q q− − q + q +q −q + q q− .

3) If 2 2 2

2 3 4

q =q +q , when q2=0, then q q= 1, and 1

1,

x qx q− _{= (2.8)}

in which x=1; When q2 ≠0, then construct a nonsingular split quaternion x

such that

1

1 i j,

x qx q− _{= + +}

_(2.9)

in which x= +1 q q3+ 4i+

(

q2−1 k

)

.

Remark 1 The statement above concludes that the set of v-quaternions is divided into four kinds of different equivalence classes. Moreover, if q q q= 1+ 2i+q3j+q4k

is a v-quaternion, then 2

(

2 2

)

1 2 3 4 i

q∈_q + q −v q +q _

  if q22 >v q

(

32+q42

)

;

2 2 2

1 3 4 1 j2

q q q q q

v

 

∈_ + + − _

  if

(

)

2 2 2

2 3 4

q <v q +q ; q∈

[ ]

q1 if q22 =v q

(

32+q24

)

and

2 0

q = _; q q₁ i 1 j

v

 

∈_ + + _

  if

(

)

2 2 2

2 3 4

q =v q +q and q2 ≠0.

3. Complex Representation of v-Quaternion Matrices

For any v-quaternion matrix A A A= 1+ 2i+A3j+A4k=

(

A A1+ 2i

)

+

(

A A3+ 4i

)

j m n

v×

∈H , where A A A A1, , ,2 3 4∈Rm n× , the complex representation of A is

de-fined to be

(

)

(

1 3 2 4

)

1 3 2 4 2 2

i i

,

i i

m n

A A v A A

A

v A A A A

σ ₌_ + + _∈ ×

− −

 

  C

 



(3.1)

in which 1, 0

1, 0

v v

>  = _{− <}



DOI: 10.4236/jamp.2019.79133 1933 Journal of Applied Mathematics and Physics It is easy to prove that the mapping Aσ _{is an isomorphism of ring} m n

v×

H to ring _C2m n×2 _{, and for ,} mn

v

A B_∈H × _, n p

v

C_∈H× _{, a∈R, we have the following}

equalities.

(

A B₊

)

σ ₌Aσ ₊Bσ,

( )

AC σ ₌A Cσ σ,

( )

aAσ ₌aAσ, _(3.2) and

T _, T _,

m n m n

Q A Qσ ₌A Q A Qσ σ ₌Aσ

(3.3)

in which 0 ₀t t

t

I Q

I

 

=  

 



, = 1,_−1,v_v>_<0₀ 

 , T

2

t t t

Q Q I= , Qt is a unitary matrix,

and clearly by (3.2) a v-quaternion matrix n n v

A_∈H× _{is nonsingular if and only}

if Aσ _{is nonsingular and}

( ) ( )

_Aσ −1_{= A}−1 σ_.

Remark 2 For two special cases of quaternion

(

v= −1

)

and split quaternion

(

v=1

)

_{matrices, clearly by (3.1) the complex representations are respectively to}

be

1 2 3 4

3 4 1 2

i i

, ,

i i

m n

A A A A

A A

A A A A

σ ₌ + − −  _∈ ×

 _{− −} 

  H (3.4a)

and

1 2 3 4

3 4 1 2

i i

, .

i i m ns

A A A A

A A

A A A A

σ ₌ + +  _∈ ×

 _{− −} 

  H (3.4b)

For any n n v

A_∈H× _{, if λ∈C is an eigenvalue of complex representation}

Aσ _{of v-quaternion matrix A, i.e. there exists a nonzero complex vector}

1 2 1

2

n

β β

β ×

  =_{ }∈

  C ,

1

1, 2 n

β β _∈C× _{, such that A}σ_{β βλ= , then by (3.3) we have}

(

n

) (

n

)

A Qσ _{β =} Q_{β λ, and}

(

, _n

) (

, _n

)

,

Aσ_{β βλ= ⇔}Aσ _β Q _{β = β} Q_{β λ}σ _⇔A_{α αλ= (3.5)}

in which 1 1 2j v

α β= + β

 , and

(

,Qn

)

σ

α = β β .

From the statement above we have the following results. Proposition 3.1 Let n n

v

A_∈H× _{, then}

1) If λ is a complex eigenvalue of complex representation Aσ_{, then λ is}

also a complex eigenvalue of complex representation Aσ_{. The imaginary}

ei-genvalue of Aσ _{appear in conjugate pairs, the number of real eigenvalues of}

Aσ _{is even.}

2) The v-quaternion matrix A has at least a complex eigenvalue. Moreover, if λ is a complex eigenvalue of Aσ _{with A}σ

_{β βλ}

_{= , let}

1 2 1 1

2

1 1 2

2

1 _j, n _{, ,} n _,

v

β

α β β β β β

β × ×

 

= + =_{ }∈ ∈

  C C



(3.6)

then λ is a right eigenvalue of the v-quaternion matrix A, and α _{is a}

DOI: 10.4236/jamp.2019.79133 1934 Journal of Applied Mathematics and Physics Proposition 3.1 gives a relation between the right complex eigenvalues of com-plex representation Aσ _{of a v-quaternion matrix A and the right complex}

eigen-values of the v-quaternion matrix A, and derives algebraic techniques for finding the right complex eigenvalues and corresponding eigenvectors of the v-quaternion matrix A.

4. Eigenvalues and Eigenvectors of v-Quaternion Matrices

For n n

v

A_∈H× _{, a nonzero v-quaternion λ, by [16], the complex representation}

matrix equation A X Xσ _{= λ}σ _{has a nonzero solution X∈C}2 2n× _{if and only if}

( )

A

f σ x and f_λσ

( )

x are not relatively prime, i.e.

(

f_Aσ

( )

x f, _λσ

( )

x

)

≠1, in

oth-er words, f_Aσ

( )

λσ is a singular matrix.

Moreover, if there exists _X∈C2 2n× _{such that A X X}σ _{= λ}σ _{, by (3.3) we have}

(

T

) (

T

)

1 1 ,

n n

A X Xσ _{= λ}σ _⇔A Q XQσ ₌ Q XQ _λσ _⇒A Y Yσ _{= λ}σ _(4.1)

in which

(

T

)

1

1

2

n

Y

=

X Q XQ

+

_{. Let}

11 12 1

21 22

, n ,

ts

X X

X X

X X

×

 

=_{ } ∈

  C

(4.2)

then it is easy to get similarly that

(

T

)

11 12 1 1

12 11

1 _{, ,}

2

n

n ts

Y Y

Y X Q XQ Y

Y Y ×

 

= + =_{ } ∈

  C (4.3)

in which

Y

11

=

1

₂

(

X

11

+

X

22

)

,

Y

12

=

1

₂

(

X

12

+



X

21

)

, 1,_1,v_v0₀

>  = _{− <}



 .

By the fact that jY11=Y11j, jY12 =Y12j, construct a v-quaternion α, we can get

the following equality by the direct calculation.

11 12

1

1 1

j , j _j ,

2 n n

Y Y I I Y

v v _v

α = + _{= }  _{ } _

 _{  }

 





 



(4.4)

then we have _{α =}σ Y_{, and the v-quaternion λ is a right eigenvalue of the}

v-quaternion matrix A and Aα αλ= _.

The statements above imply following result. Theorem 4.1 Let n n,

v v

A_∈H×

λ

_{∈H . Then}

1) The v-quaternion λ is a right eigenvalue of the v-quaternion matrix A if and only if

(

f_Aσ

( )

x f, _λσ

( )

x

)

≠1, i.e. f_Aσ

( )

λσ is a singular complex matrix;

2) The v-quaternion λ is a right eigenvalue of A if and only if complex re-presentation matrix equation A Y Yσ _{= λ}σ _{has a nonzero solution X∈C}2 2n× _.

In which case, if _X∈C2 2n× _{is a nonzero solution of A X X}σ _{= λ}σ _{, let}

(

T

)

1 1

1 _, 1 _{j ,}

j 4 In In X Q XQn

v _v

α

 

  _{ }

= _{ } + _{ }

  _{ }

 

 



(4.5)

in which 0

0

t t

t

I Q

I

 

=  

 



, 1, 0

1, 0

v v

>  = _{− <}



 , then nonzero vector n1

v

DOI: 10.4236/jamp.2019.79133 1935 Journal of Applied Mathematics and Physics

Aα αλ= . That is α is an eigenvector of the v-quaternion matrix A related to the right eigenvalue λ.

This section researches the right eigenvalues and eigenvectors of a v-quaternion matrix with the help of the complex representation of the v-quaternion matrix, in this way, it makes the eigen-problem of the v-quaternion into a new topics, it is the matrix equation problem of complex representation. It also derives an al-gebraic technique to finding the right eigenvalues and corresponding eigenvec-tors of the v-quaternion.

5. Algebraic Techniques for Eigenvalues and Eigenvectors

For n n

v

A_∈H× _{, let V}

{

2 1n _|Aσ

}

λ = β∈C × β βλ= be the characteristic subspace of

complex eigenvalue λ. By Proposition 3.1, let all complex eigenvalues of com-plex representation Aσ _{be as follows.}

1, , , , , , , ,1 t t 1 2 2s,

λ λ  λ λ µ µ  µ

in which λµ are imaginary and µv are real, 1< <µ t and 1< <v 2s.

Case 1: For an imaginary eigenvalue λ λ= u of Aσ, let Aσ

β βλ β

= , ≠0, by (3.3)

(

, n

) (

, n

)

.

Aσ_{β βλ= ⇔}Aσ _β Q_{β = β} Q_{β λ}σ _(5.1)

Then by (4.5) construct the nonzero vector n1

v

H

α_∈ × _{satisfies Aα αλ= .}

That is α _{is an eigenvector of the v-quaternion matrix A related to the right}

eigenvalue λ.

Case 2: For a real eigenvalue λ µ= u of Aσ, if there exist more than one dif-ferent real eigenvalues, let λ µ= u and µ µ= v be two different real eigenva-lues, Aσ

_{β βλ}

₌ ,Aσ

_{γ γµ}

_{= , then}

(

,

) (

,

)

0 ,

0

Aσ _{β γ β γ} λ

µ

 

= _{ }

  (5.2)

therefore there exists non-negative real number v>0 such that

(

,

)

(

,

)

2 2 2 2

2 2 2 2

j ,

2 2

A P P A Y Y

A Y Y A Y Y

v

σ σ

σ

σ σ σ

λ µ λ µ λ µ λ µ

β γ β γ

λ µ λ µ λ µ λ µ

λ µ λ µ _η

+ − + −

   

   

=  ⇔ =  

− + − +

   

   

   

+ −

 

⇔ = _ + _ ⇔ =

 

(5.3)

in which 1 1

1 1

P_{= } _

−

 , Y =

(

β γ

,

)

P, 2 2 v j

λ µ λ µ

η= + + − . Then by (4.5)

con-struct the n1

v

α_∈H× _{satisfies j}

2 2

A

v

λ µ λ µ α α= _ + + − _

 . It is easy to prove that α is a nonzero vector by the fact that β and γ are linearly independent. That is α _{is an eigenvector of the v-quaternion matrix A related to the right}

eigenvalue j

2 2 v

DOI: 10.4236/jamp.2019.79133 1936 Journal of Applied Mathematics and Physics Case 3: For real eigenvalue λ of Aσ_{, if there exists only one real eigenvalue}

λ and dimVλ =1, there exists a Jordan block J

( )

λ

of real eigenvalue λ

and a column full rank matrix Y such that A Y YJσ ₌

( )

_λ

_{, then there exists two}

linearly independent vectors β γ, _{such that}

(

,

) (

,

)

1 0

Aσ _{β γ β γ} λ

λ

 

= _{ }

 , and

clearly

(

,

) (

,

)

1 i 1 ,

0 1 i

Aσ _{β γ β γ} λ A Y Yσ λ

λ λ

+

   

= _{ }⇔ = _{ }

−

    (5.4)

and clear that 1 i 1 1

0 P 1 i P

λ λ

λ λ −

+

   

=

   ₋ 

    , in which

1 0 i 1 P_{= } _

 .

Therefore there only exists non-negative real number v>0 such that

(

)

(

)

i 1

1

i j , , ,

1 i

A Y Y A P P

v

σ _λ σ _{β γ β γ} λ

λ +

 

 

= _ + + _⇔ = _{ }

−

    (5.5)

in which

(

,

)

(

,

)

1 0

(

i,

)

i 1

Y = β γ P= β γ _ _= β γ γ+

  . Then by (4.5) construct the

1

n v

α

_∈H× _{satisfies A i} 1 _j

v

α α λ= _ + + _

 . It is easy to prove that α is a

non-zero vector by the fact that β _and γ are linearly independent. That is α _is

an eigenvector of the v-quaternion matrix A related to the right eigenvalue 1

i j

v

λ+ + .

Case 4: For real eigenvalue λ of Aσ_{, If there exists only one real eigenvalue}

λ of Aσ _{, and dimV 2}

λ ≥ , there exist two linearly independent vectors

, Vλ

β γ ∈ such that Aσ

_{β βλ}

₌ ,Aσ

_{γ γλ}

_{= , then}

(

) (

)

0

, , , ,

0

Aσ_{β βλ} Aσ_{γ γλ} Aσ _{β γ β γ} λ A Y Yσ _λσ

λ

 

= = ⇔ = _{ }⇔ =

  (5.6)

in which Y =

(

β γ,

)

. Then by (4.5) construct the n1

v

α

_∈H× _satisfies

Aα αλ= , and α _{is a nonzero vector by the fact that} β _and γ are linearly indepen-dent. That is α is an eigenvector of the v-quaternion matrix A related to the right eigenvalue λ.

For n n v

A_∈H× _{, if a v-quaternion λ is a right eigenvalue of A, and}

Aα αλ= , then for any nonsingular v-quaternion p, we have _{A p}

( ) ( )

_{α = αp p}

(

−1_λp

)

_{, i.e.}

1

p−

λ

p _{is a right eigenvalue of A with an eigenvector}

p

α .

6. Algorithm and Example Algorithm

Let n n

v

A_∈H× _{. Then an algorithm for right v-quaternion eigenvalues and}

eigen-vectors of the v-quaternion matrix is given as follows. 1) Compute the complex representation Aσ _{of A by (3.1).}

2) Compute all eigenvalues of Aσ_{. Let all the eigenvalues of A}σ _{be as}

fol-lows.

1, , , , , , , ,1 t t 1 2 2s,

DOI: 10.4236/jamp.2019.79133 1937 Journal of Applied Mathematics and Physics in which λµ are imaginary and µv are real, 1< <µ t and 1< <v 2s.

3) Choose one of each pair of conjugate complex eigenvalues and all the real eigenvalues of Aσ

. Then can get the related eigenvalues of A.

4) Search the eigenvectors related to eigenvalues of the Aσ _{in step 3.}

5) If λ _{is an eigenvalue of the v-quaternion matrix A, then it is easy to get a}

2 2n

Y_∈C × _{such that A Y Y}σ _{= λ}σ _{by finding a solution of complex matrix}

equa-tion A X Xσ _{= λ}σ _{. Let}

(

T

)

1 1

1 _, 1 _{j ,}

j 4 In In Y Q YQn

v

v

α

 

  _{ }

= _{ } + _{ }

  _{ }

 

 



(6.1)

then nonsingular vector n1

v

α_∈H× _satisfies

Aα αλ= . That is α is an eigen-vector of the v-quaternion matrix A related to the right eigenvalue λ.

6) Find out the equivalence classes of all the eigenvalues, find the related ei-genvectors of v-quaternion A.

Example Let

2 i 2j 2 4j 2k .

i j 1 i 2j k

A_{= } + − − − _

− + − + +

 

The complex representation of the v-quaternion matrix A is as follows

(

)

(

)

2 i 2 2 2 2

i 1 i 2 _,

2 4 2 2 i 2

2 i 1 i

v v v

v v v

A

v v v

v v v

σ

 _{+ − − + −} 

 

 _{− − + −} 

 

=

_{− − + − −} 

 

 _{− − +} 

 

    

    

  

  

in which 1, 0

1, 0

v v

>  = _{− <}



 .

The following two examples are special cases of the v-quaternion.

Case 1: For quaternion case with

(

v= −1

)

. It is easy to know Aσ _{by (3.4a),}

2 i 2 2 4 2i

i 1 i 1 2 i

,

2 4 2i 2 i 2

1 2 i i 1 i

Aσ

+ +

 

 _{− − − − −} 

 

=

− − + − 

 _{− +} 

 

by the direct calculation get the eigenvalues of Aσ _{are 2 i} µ

λ = + ,

λ

µ = −2 i,

1 2i

η

λ

= + ,

λ

η = −1 2i.

Choose two eigenvalues λ = +µ 2 i,

λ

η = +1 2i of two pairs of conjugate

eigenvalues.

For the eigenvalue λ = +µ 2 i. It is easy to get that Aσβ βλ= , in which

(

)

T

2, 1, 1 i,1

β = − − − . By (4.5), let

(

)

(

(

)

T

(

)

)

2 2 2 2 2 1

1 2 j k

1 _{, j , , ,}

j 1 j

4 I I Q Q Q Q

α = β β + β β   _{  }= − − _

− − +

   

DOI: 10.4236/jamp.2019.79133 1938 Journal of Applied Mathematics and Physics nonsingular quaternion p, quaternion _p 1 _p

µ

λ

−

is a right eigenvalue of A with an eigenvector αp.

In the same way, for the eigenvalue

λ

η = +1 2i. It is easy to get that Aσ

η

β βλ

= _{, in which} β =

(

2 2 2, 1+ − − 2, 2,1−

)

T. By (4.5), let

(

)

(

(

)

T

(

)

)

(

)

2 2 2 2 2 1

2 1 2 j

1

1 _{, j , , ,}

j

4 I I Q Q Q Q _{1 2 j}

α = β β + β β   _{ }= + − 

−  

  _{ − − + }

then Aα αλ= η, α is an eigenvector related to

λ

η = +1 2i. Moreover, for any

nonsingular quaternion p, quaternion _p1 _p

η

λ

−

is a right eigenvalue of A with an eigenvector αp.

Case 2: For split quaternion case with

(

v=1

)

. It is easy to know Aσ _by

(3.4b),

2 i 2 2 4 2i

i 1 i 1 2 i

,

2 4 2i 2 i 2

1 2 i i 1 i

Aσ

+ − − −

 

 _{− − +} 

 

=

− − + − 

 _{− +} 

 

it is easy to get the eigenvalues of Aσ _{are λ= +2 i, λ = −2 i, µ=1, and}

dimVµ =1.

For two conjugate imaginary eigenvalues λ= +2 i, λ = −2 i, choose an ei-genvalue λ= +2 i of a pairs of conjugate eigenvalues. For the only one real ei-genvalue µ=1 with dimVµ =1, by (5.6) get an eigenvalue λ = + +1 i j of A.

For the principal eigenvalue λ= +2 i, It is easy to get that Aσ

_{β βλ}

_{= , in}

which _{β =}

(

_{2, 1, 1,0− −}

)

T_{. By (4.5), let}

(

)

(

(

)

T

(

)

)

2 2 2 2 2 1

1 2 j

1 _{, j , , ,}

j 1

4 I I Q Q Q Q

α = β β + β β   _{  }= − _

−

   

then Aα αλ= _, α _{is an eigenvector related to} λ= +2 i_{. Moreover, for any} nonsingular split quaternion p, split quaternion _p−1

_λ

_{p is a right eigenvalue of}

A with an eigenvector αp.

For the principal eigenvalue µ=1 _with dimVµ =1. It is easy to get two

li-nearly independent vectors β γ, _{such that}

(

,

) (

,

)

1 ,

0

Aσ _{β γ β γ} µ

µ

 

= _{ }

 

in which _{β =}

(

_{2, 1, 2,1− −}

)

T_{, i, , i,}i i T

2 2

γ = −_ − _

  . By (4.5), let

(

) (

(

)

T

(

)

)

2 2 2 1

3 i j k 1

1 _{, j i, i, ,}

3 1 1 1

j

4 i j k

2 2 2 2

I I Q Q

α β γ γ β γ γ

+ − −

 

   

= + + − _{  }=



− − + +

   

then Aα αλ= , α is an eigenvector related to λ µ= + + = + +i j 1 i j. Moreo-ver, for any nonsingular split quaternion p, split quaternion _p−1

_λ

_{p is a right}

DOI: 10.4236/jamp.2019.79133 1939 Journal of Applied Mathematics and Physics

7. Conclusion

This paper gives the concept of the v-quaternions, and studies eigenvalues and eigenvectors of the v-quaternion matrices by means of the complex representa-tion of the v-quaternion matrices, and derives an algebraic technique to find the eigenvalues and eigenvectors of v-quaternion matrices. This paper also gives a unification of algebraic techniques for eigenvalues and eigenvectors in quater-nionic and split quaterquater-nionic mechanics.

Acknowledgements

We thank the Editor and the referee for their comments. Research of T. Jiang is funded by the National Natural Science Foundation of China (11771188) and Shandong Natural Science Foundation (ZR201709250116). This support is greatly appreciated.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this pa-per.

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