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This paper aims to present, in a unified manner, algebraic techniques for linear equations which are valid on both the algebras of quaternions and split quaternions. This paper, introduces a concept of v-quaternion, studies the problem of v-quaternionic linear equations by means of a complex representation and a real representation of v-quaternion matrices, and gives two algebraic methods for solving v-quaternionic linear equations. This paper also gives a unification of algebraic techniques for quaternionic and split quaternionic linear equations in quaternionic and split quaternionic mechanics.

A quaternion, which was found in 1840 by William Rowan Hamilton [

A split quaternion (or coquaternion), which was found in 1849 by James Cockle [

A v-quaternion is in the form of

q = q 1 + q 2 i + q 3 j + q 4 k , i 2 = − 1, j 2 = v , ij = − ji = k , (1.1)

in which 0 ≠ v ∈ R , q 1 , q 2 , q 3 , q 4 ∈ R , and k 2 = ijk = v , jk = − kj = − v i , ik = − ki = − j . Let H v denote the set of v-quaternion. Obviously, the set of all v-quaternion is also a noncommutative 4-dimensional Clifford algebra. Specially, when v = − 1 , the ring of the v-quaternion H v is the ring of the quaternion H ; when v = 1 , the ring of the v-quaternion H v is the ring of the split quaternion H s .

In the geometry research and physical application of quaternion and split quaternion, the problems of solving quaternionic and split quaternionic equations are often encountered. In paper [

Let R be the real number field, C = R ⊕ R i the complex number field. If q = q 1 + q 2 i + q 3 j + q 4 k ∈ Η v , q ¯ = q 1 − q 2 i − q 3 j − q 4 k is the conjugate of q. For any matrix A = ( a s t ) ∈ H v m × n , A ¯ = ( a ¯ s t ) , A T = ( a t s ) , A * = ( a ¯ t s ) , A − 1 denote the conjugate, the transpose, the conjugate transpose and the inverse of the matrix A, respectively.

This paper is organized as follows. In Section 2, we give two new matrix representations of v-quaternion matrix, and discuss some properties and conclusions of complex representation and real representation of v-quaternion matrices. In Section 3, we present the complex representation method for solving v-quaternionic linear equations and some numerical examples. In Section 4, we present the real representation method for solving v-quaternionic linear equations and some numerical examples. In Section 5, we summarize this paper.

For any v-quaternion matrix A = A 1 + A 2 i + A 3 j + A 4 k = ( A 1 + A 2 i ) + ( A 3 + A 4 i ) j = B 1 + B 2 j ∈ H v m × n , A 1 , A 2 , A 3 , A 4 ∈ R m × n , B 1 , B 2 ∈ C m × n , the complex representation A C of the v-quaternion matrix A is defined to be

A C = [ B 1 v B 2 B ¯ 2 B ¯ 1 ] , (2.1)

and the real representation A R of the v-quaternion matrix A is defined to be

A R = [ A 1 − A 2 v A 3 v A 4 A 2 A 1 v A 4 − v A 3 A 3 A 4 A 1 − A 2 A 4 − A 3 A 2 A 1 ] . (2.2)

For any v-quaternion matrix A , B ∈ H v m × n , C ∈ H v n × p , a ∈ R , for σ ∈ { C , R } , it is easy to prove the following equalities by direct calculation.

( A + B ) σ = A σ + B σ , ( a A ) σ = a A σ , ( A C ) σ = A σ C σ , (2.3)

and

Q m − 1 A C Q n = A C ¯ , (2.4)

where Q t = [ 0 v I t I t 0 ] .

Similarly, by direct calculation we get the following results.

P m − 1 A R P n = A R , R m − 1 A R R n = A R , S m − 1 A R S n = A R , (2.5)

where P t = [ 0 − I t 0 0 I t 0 0 0 0 0 0 I t 0 0 − I t 0 ] , R t = [ 0 0 v I t 0 0 0 0 v I t I t 0 0 0 0 I t 0 0 ] , S t = [ 0 0 0 v I t 0 0 − v I t 0 0 − I t 0 0 I t 0 0 0 ] , and P t − 1 = − P t , R t − 1 = 1 v R t , S t − 1 = 1 v S t .

Lemma 2.1 For two special cases of quaternion ( v = − 1 ) and split quaternion ( v = 1 ) matrices, clearly by (2.1) and (2.2) the complex representation and the real representation are respectively to be

A C = [ B 1 − B 2 B ¯ 2 B ¯ 1 ] , A R = [ A 1 − A 2 − A 3 − A 4 A 2 A 1 − A 4 A 3 A 3 A 4 A 1 − A 2 A 4 − A 3 A 2 A 1 ] , A ∈ H m × n . (2.6a)

A C = [ B 1 B 2 B ¯ 2 B ¯ 1 ] , A R = [ A 1 − A 2 A 3 A 4 A 2 A 1 A 4 − A 3 A 3 A 4 A 1 − A 2 A 4 − A 3 A 2 A 1 ] , A ∈ H s m × n . (2.6b)

For any v-quaternion matrix A ∈ H v m × n , the rank rank ( A ) of the matrix A is defined to be

rank ( A ) ≡ 1 2 rank ( A C ) , (2.7)

or

rank ( A ) ≡ 1 4 rank ( A R ) . (2.8)

By the definition of rank and (2.3), it is easy to get the following results by direct calculation. If A , B ∈ Η v m × n , C ∈ Η v n × p , then rank ( A + B ) ≤ rank ( A ) + rank ( B ) and rank ( A C ) ≤ min { rank ( A ) , rank ( C ) } .

If A ∈ H v m × n , B ∈ H v m × p , then by the definition of complex representation and (2.3), A X = B if and only if A C X C = B C . That is A X = B has a solution X if and only if A C Y = B C has a solution Y = X C .

Theorem 3.1 For A ∈ H v m × n , B ∈ H v m × p . Then

1) V-quaternionic linear equations A X = B have a solution if and only if rank ( A ) = rank ( A , B ) , i.e. A X = B has a solution if and only if A C Y = B C has a solution, and if rank ( A ) = rank ( A , B ) = n , then v-quaternionic linear equations A X = B have a unique solution.

2) If Y is a solution to A C Y = B C , then the following v-quaternion matrix is a solution to A X = B ,

X = 1 4 [ I n , I n j ] ( Y + Q n − 1 Y ¯ Q p ) [ I p 1 v I p j ] , (3.1)

in which Q t = [ 0 v I t I t 0 ] .

Proof: If Y is a solution of A C Y = B C , by (2.4),

A C Y = B C ⇔ A C ¯ ( Q n − 1 Y Q p ) = B C ¯ ⇔ A C ( Q n − 1 Y ¯ Q p ) = B C , (3.2)

i.e. Q n − 1 Y ¯ Q p is a solution of A C Y = B C , therefore

Y ^ = 1 2 ( Y + Q n − 1 Y ¯ Q p ) (3.3)

is also a solution of A C Y = B C . Let

Y = [ z 11 z 12 z 21 z 22 ] ∈ C 2 n × 2 p , z t s ∈ C n × p , s , t = 1 , 2. (3.4)

It is easy to get, by direct calculation,

Y ^ = [ z ^ 1 v z ^ 2 z ^ ¯ 2 z ^ ¯ 1 ] ∈ C 2 n × 2 p , (3.5)

in which

z ^ 1 = 1 2 ( z 11 + z ¯ 22 ) , z ^ 2 = 1 2 ( 1 v z 12 + z ¯ 21 ) . (3.6)

By (3.5), we construct a v-quaternion matrix.

X = z ^ 1 + z ^ 2 j = 1 2 [ I n , I n j ] Y ^ [ I p 1 v I p j ] . (3.7)

Clearly X C = Y ^ . This means that X C = Y ^ is a solution of A C Y = B C , so X is a solution of A X = B .

From the statement above we get following results. When the v-quaternionic linear equations A X = B have a solution, we can find a solution by a solution of complex representation equation A C Y = B C from the formula (3.1).

The following two special cases about quaternions and split quaternions come from Theorem 3.1 respectively with v = − 1 and v = 1 .

Corollary 3.2 For A ∈ H m × n , B ∈ H m × p . Then

1) The quaternionic linear equations A X = B have a solution if and only if rank ( A ) = rank ( A , B ) , i.e. A X = B has a solution if and only if A C Y = B C has a solution, and if rank ( A ) = rank ( A , B ) = n , then quaternionic linear equations A X = B have a unique solution.

2) If Y is a solution to A C Y = B C , then the following quaternion matrix is a solution to A X = B ,

X = 1 4 [ I n , I n j ] ( Y + Q n − 1 Y ¯ Q p ) [ I p − I p j ] , (3.8)

in which Q t = [ 0 − I t I t 0 ] .

Corollary 3.3 For A ∈ H s m × n , B ∈ H s m × p . Then

1) The split quaternionic linear equations A X = B have a solution if and only if rank ( A ) = rank ( A , B ) , i.e. A X = B has a solution if and only if A C Y = B C has a solution, and if rank ( A ) = rank ( A , B ) = n , then split quaternionic linear equations A X = B have a unique solution.

2) If Y is a solution to A C Y = B C , then the following split quaternion matrix is a solution to A X = B ,

X = 1 4 [ I n , I n j ] ( Y + Q n − 1 Y ¯ Q p ) [ I p I p j ] , (3.9)

in which Q t = [ 0 I t I t 0 ] .

In the similarly way, we have the following result.

Theorem 3.4 For A ∈ H v m × n , C ∈ H v p × q , B ∈ H v m × q . Then

1) V-quaternionic matrix equation A X C = B has a solution if and only if rank ( A ) = rank ( A , B ) and rank ( C ) = rank ( C B ) , i.e. A X C = B has a solution if and only if A C Y C C = B C has a solution, and if rank ( A ) = rank ( A , B ) = n , rank ( C ) = rank ( C B ) = p , then v-quaternionic matrix equation A X C = B has a unique solution.

2) If Y is a solution to A C Y C C = B C , then the following v-quaternion matrix is a solution to A X C = B ,

X = 1 4 [ I n , I n j ] ( Y + Q n − 1 Y ¯ Q p ) [ I p 1 v I p j ] , (3.10)

in which Q t = [ 0 v I t I t 0 ] .

The proof process is similar to the Theorem 3.1.

Remark 1 The above theorems and corollaries not only give the necessary and sufficient conditions for quaternion and split quaternion matrix equations A X = B , A X C = B to have a solution, but also a unification of representation for a solution.

Example 3.1

Let

A = ( i 1 + j − 1 + j − k ) and B = ( i − 1 ) .

Find all solutions of the v-quaternionic linear equations A X = B .

By the complex representation of the v-quaternion matrix, we know

A C = [ i 1 0 v − 1 0 v − v i 0 1 − i 1 1 i − 1 0 ] , B C = [ i 0 − 1 0 0 − i 0 − 1 ] ,

and if v ≠ 7 ± 45 2 , then rank ( A C ) = rank ( A C , B C ) = 4 , i.e. rank ( A ) = rank ( A , B ) = 2 , then the v-quaternionic linear equations A X = B have a unique solution.

For the matrix equation A C Y = B C , the unique solution is easily found to be

Y = [ − 5 v + 1 v 2 − 7 v + 1 v ( 1 − 2 v ) v 2 − 7 v + 1 − 3 v v 2 − 7 v + 1 i v ( − 1 − v ) v 2 − 7 v + 1 i 1 − 2 v v 2 − 7 v + 1 − 5 v + 1 v 2 − 7 v + 1 1 + v v 2 − 7 v + 1 i 3 v v 2 − 7 v + 1 i ] .

By (3.1), we easily find the unique solution X of v-quaternionic linear equations A X = B , and

X = 1 4 [ I 2 , I 2 j ] ( Y + Q n − 1 Y ¯ Q p ) [ 1 1 v j ] = [ − 5 v + 1 v 2 − 7 v + 1 + 1 − 2 v v 2 − 7 v + 1 j − 3 v v 2 − 7 v + 1 i − 1 + v v 2 − 7 v + 1 k ] T .

The following two examples are special cases of the above conclusion.

Case 1: For quaternionic linear equations A X = B with v = − 1 . It is easy to know A C and B C by (2.6a),

A C = [ i 1 0 − 1 − 1 0 − 1 i 0 1 − i 1 1 i − 1 0 ] , B C = [ i 0 − 1 0 0 − i 0 − 1 ]

and rank ( A C ) = rank ( A C , B C ) = 4 . Clearly, the linear equations A C Y = B C have a unique solution. The unique solution is easily found to be

Y = [ 2 3 − 1 3 1 3 i 0 1 3 2 3 0 − 1 3 i ] .

By (3.8), we easily find the unique solution X of quaternionic linear equations A X = B , and

X = 1 4 [ I 2 , I 2 j ] ( Y + Q n − 1 Y ¯ Q p ) [ 1 − j ] = [ 2 3 + 1 3 j 1 3 i ] T .

Case 2: For split quaternionic linear equations A X = B with v = 1 . It is easy to know A C and B C by (2.6b),

A C = [ i 1 0 1 − 1 0 1 − i 0 1 − i 1 1 i − 1 0 ] , B C = [ i 0 − 1 0 0 − i 0 − 1 ]

and rank ( A C ) = rank ( A C , B C ) = 4 . Clearly, the linear equations A C Y = B C have a unique solution. The unique solution is easily found to be

Y = [ 4 5 1 5 3 5 i 2 5 i 1 5 4 5 − 2 5 i − 3 5 i ] .

By (3.9), we easily find the unique solution X of split quaternionic linear equations A X = B , and

X = 1 4 [ I 2 , I 2 j ] ( Y + Q n − 1 Y ¯ Q p ) [ 1 j ] = [ 4 5 + 1 5 j 3 5 i + 2 5 k ] T .

If A ∈ H v m × n , B ∈ H v m × p , then by the definition of real representation, A X = B if and only if A R X R = B R . That is A X = B has a solution X if and only if A R Y = B R has a solution Y = X R .

Theorem 4.1 For A ∈ H v m × n , B ∈ H v m × p . Then

1) V-quaternionic linear equations A X = B have a solution if and only if rank ( A ) = rank ( A , B ) , i.e. A X = B has a solution if and only if A R Y = B R has a solution, and if rank ( A ) = rank ( A , B ) = n , then v-quaternionic linear equations A X = B have a unique solution.

2) If Y is a solution to A R Y = B R , then the following v-quaternion matrix is a solution to A X = B ,

X = 1 16 [ I n , I n i , I n j , I n k ] ( Y − P n Y P p + 1 v R n Y R p + 1 v S n Y S p ) [ I p − I p i 1 v I p j 1 v I p k ] . (4.1)

Proof: If Y is a solution of A R Y = B R , by (2.5),

A R Y = B R ⇔ P m − 1 A R P n ( P n − 1 Y P p ) = P m − 1 B R P p ⇔ A R ( P n − 1 Y P p ) = B R , (4.2)

i.e. P n − 1 Y P p is a solution of A R Y = B R . Similarly, R n − 1 Y R p , S n − 1 Y S p are also solution of A R Y = B R .

Y ^ = 1 4 ( Y + P n − 1 Y P p + R n − 1 Y R p + S n − 1 Y S p ) = 1 4 ( Y − P n Y P p + 1 v R n Y R p + 1 v S n Y S p ) (4.3)

is also a solution of A R Y = B R . Let

Y = [ z 11 z 12 z 13 z 14 z 21 z 22 z 23 z 24 z 31 z 32 z 33 z 34 z 41 z 42 z 43 z 44 ] ∈ R 4 n × 4 p , z t s ∈ R n × p , s , t = 1 , 2 , 3 , 4. (4.4)

It is easy to get, by direct calculation,

Y ^ = [ z ^ 1 − z ^ 2 v z ^ 3 v z ^ 4 z ^ 2 z ^ 1 v z ^ 4 − v z ^ 3 z ^ 3 z ^ 4 z ^ 1 − z ^ 2 z ^ 4 − z ^ 3 z ^ 2 z ^ 1 ] ∈ R 4 n × 4 p , (4.5)

in which

z ^ 1 = 1 4 ( z 11 + z 22 + z 33 + z 44 ) , z ^ 2 = 1 4 ( z 21 − z 12 + z 43 − z 34 ) , (4.6a)

z ^ 3 = 1 4 ( z 31 − z 42 + 1 v z 13 − 1 v z 24 ) , z ^ 4 = 1 4 ( z 41 + z 32 + 1 v z 23 + 1 v z 14 ) . (4.6b)

By (4.5), we construct a v-quaternion matrix.

X = z ^ 1 + z ^ 2 i + z ^ 3 j + z ^ 4 k = 1 4 [ I n , I n i , I n j , I n k ] Y ^ [ I p − I p i 1 v I p j 1 v I p k ] . (4.7)

Clearly X R = Y ^ . This means that X R = Y ^ is a solution of A R Y = B R , so X is a solution of A X = B .

From the statement above we get following results.

The following two special cases about quaternions and split quaternions come from Theorem 4.1 respectively with v = − 1 and v = 1 .

Corollary 4.2 For A ∈ H m × n , B ∈ H m × p . Then

1) The quaternionic linear equations A X = B have a solution if and only if rank ( A ) = rank ( A , B ) , i.e. A X = B has a solution if and only if A R Y = B R has a solution, and if rank ( A ) = rank ( A , B ) = n , then quaternionic linear equations A X = B have a unique solution.

2) If Y is a solution to A R Y = B R , then the following quaternion matrix is a solution to A X = B ,

X = 1 16 [ I n , I n i , I n j , I n k ] ( Y − P n Y P p − R n Y R p − S n Y S p ) [ I p − I p i − I p j − I p k ] . (4.8)

Corollary 4.3 For A ∈ H s m × n , B ∈ H s m × p . Then

1) The split quaternionic linear equations A X = B have a solution if and only if rank ( A ) = rank ( A , B ) , i.e. A X = B has a solution if and only if A R Y = B R has a solution, and if rank ( A ) = rank ( A , B ) = n , then split quaternionic linear equations A X = B have a unique solution.

2) If Y is a solution to A R Y = B R , then the following split quaternion matrix is a solution to A X = B ,

X = 1 16 [ I n , I n i , I n j , I n k ] ( Y − P n Y P p + R n Y R p + S n Y S p ) [ I p − I p i I p j I p k ] . (4.9)

In the similarly way, we have the following result.

Theorem 4.4 For A ∈ H v m × n , C ∈ H v p × q , B ∈ H v m × q . Then

1) V-quaternionic matrix equation A X C = B has a solution if and only if rank ( A ) = rank ( A , B ) and rank ( C ) = rank ( C B ) , i.e. A X C = B has a solution if and only if A R Y C R = B R has a solution, and if rank ( A ) = rank ( A , B ) = n , rank ( C ) = rank ( C B ) = p , then v-quaternionic matrix equation A X C = B has a unique solution.

2) If Y is a solution to A R Y C R = B R , then the following v-quaternion matrix is a solution to A X C = B ,

X = 1 16 [ I n , I n i , I n j , I n k ] ( Y − P n Y P p + 1 v R n Y R p + 1 v S n Y S p ) [ I p − I p i 1 v I p j 1 v I p k ] . (4.10)

The proof process is similar to the Theorem 4.1.

Remark 2 The above theorems and corollaries not only give the necessary and sufficient conditions for quaternion and split quaternion matrix equations A X = B , A X C = B to have a solution, but also a unification of representation for a solution.

Example 4.1

For two v-quaternion matrices A and B in Example 3.1, find solutions of the v-quaternionic linear equations A X = B .

By the real representation of the v-quaternion matrix, we know

A R = [ 0 1 − 1 0 0 v 0 0 − 1 0 0 0 v 0 0 − v 1 0 0 1 0 0 0 − v 0 0 − 1 0 0 − v − v 0 0 1 0 0 0 1 − 1 0 1 0 0 − 1 − 1 0 0 0 0 0 0 − 1 1 0 0 1 0 − 1 − 1 0 0 0 − 1 0 ] , B R = [ 0 − 1 0 0 − 1 0 0 0 1 0 0 0 0 − 1 0 0 0 0 0 − 1 0 0 − 1 0 0 0 1 0 0 0 0 − 1 ] ,

and if v ≠ 7 ± 45 2 , then rank ( A R ) = rank ( A R , B R ) = 8 , i.e. rank ( A ) = rank ( A , B ) = 2 , then the v-quaternionic linear equations A X = B have a unique solution.

For the matrix equation A R Y = B R , the unique solution is easily found to be

Y = 1 v 2 − 7 v + 1 [ − 5 v + 1 0 ( 1 − 2 v ) v 0 0 3 v 0 ( − 1 − v ) v 0 − 5 v + 1 0 ( 2 v − 1 ) v − 3 v 0 ( − 1 − v ) v 0 1 − 2 v 0 − 5 v + 1 0 0 − 1 − v 0 3 v 0 2 v − 1 0 − 5 v + 1 − 1 − v 0 − 3 v 0 ] .

By (4.1), we easily find the unique solution X of v-quaternionic linear equations A X = B , and

X = 1 16 [ I 2 , I 2 i , I 2 j , I 2 k ] ( Y − P n Y P p + 1 v R n Y R p + 1 v S n Y S p ) [ 1 − i 1 v j 1 v k ] = [ − 5 v + 1 v 2 − 7 v + 1 + 1 − 2 v v 2 − 7 v + 1 j − 3 v v 2 − 7 v + 1 i − 1 + v v 2 − 7 v + 1 k ] T .

The following two examples are special cases of the above conclusion.

Case 1: For quaternionic linear equations A X = B with v = − 1 . It is easy to know A R and B R by (2.6a),

A R = [ 0 1 − 1 0 0 − 1 0 0 − 1 0 0 0 − 1 0 0 1 1 0 0 1 0 0 0 1 0 0 − 1 0 0 1 1 0 0 1 0 0 0 1 − 1 0 1 0 0 − 1 − 1 0 0 0 0 0 0 − 1 1 0 0 1 0 − 1 − 1 0 0 0 − 1 0 ] , B R = [ 0 − 1 0 0 − 1 0 0 0 1 0 0 0 0 − 1 0 0 0 0 0 − 1 0 0 − 1 0 0 0 1 0 0 0 0 − 1 ] ,

and rank ( A R ) = rank ( A R , B C ) = 8 . Clearly, the linear equations A R Y = B R have a unique solution. The unique solution is easily found to be

Y = 1 9 [ 6 0 − 3 0 0 − 3 0 0 0 6 0 3 3 0 0 0 3 0 6 0 0 0 0 − 3 0 − 3 0 6 0 0 3 0 ] .

By (4.8), we easily find the unique solution X of quaternionic linear equations A X = B , and

X = 1 16 [ I 2 , I 2 i , I 2 j , I 2 k ] ( Y − P n Y P p − R n Y R p − S n Y S p ) [ 1 − i − j − k ] = [ 2 3 + 1 3 j 1 3 i ] T .

Case 2: For split quaternionic linear equations A X = B with v = 1 . It is easy to know A R and B R by (2.6b),

A R = [ 0 1 − 1 0 0 1 0 0 − 1 0 0 0 1 0 0 − 1 1 0 0 1 0 0 0 − 1 0 0 − 1 0 0 − 1 − 1 0 0 1 0 0 0 1 − 1 0 1 0 0 − 1 − 1 0 0 0 0 0 0 − 1 1 0 0 1 0 − 1 − 1 0 0 0 − 1 0 ] , B R = [ 0 − 1 0 0 − 1 0 0 0 1 0 0 0 0 − 1 0 0 0 0 0 − 1 0 0 − 1 0 0 0 1 0 0 0 0 − 1 ]

and rank ( A R ) = rank ( A R , B R ) = 8 . Clearly, the linear equations A R Y = B R have a unique solution. The unique solution is easily found to be

Y = − 1 5 [ − 4 0 − 1 0 0 3 0 − 2 0 − 4 0 1 − 3 0 − 2 0 − 1 0 − 4 0 0 − 2 0 3 0 1 0 − 4 − 2 0 − 3 0 ] .

By (4.9), we easily find the unique solution X of split quaternionic linear equations A X = B , and

X = 1 16 [ I 2 , I 2 i , I 2 j , I 2 k ] ( Y − P n Y P p + R n Y R p + S n Y S p ) [ 1 − i j k ] = [ 4 5 + 1 5 j 3 5 i + 2 5 k ] T .

The goal of this paper is to solve the quaternion and split quaternion linear equations in a unified manner. First, we give the definition of the v-quaternion and two new matrix representations of v-quaternion matrix. Then we derive two algebraic methods for solving the linear equations of v-quaternion. It is noteworthy that this paper not only gives algebraic techniques for solving the linear equations over v-quaternion algebras, but also a unification of algebraic techniques for linear equations in quaternionic and split quaternionic theory.

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.

The authors declare no conflicts of interest regarding the publication of this paper.

Wang, G., Guo, Z.W., Zhang, D. and Jiang, T.S. (2019) Algebraic Techniques for Linear Equations over Quaternions and Split Quaternions: A Unified Approach in Quaternionic and Split Quaternionic Mechanics. Journal of Applied Mathematics and Physics, 7, 1718-1731. https://doi.org/10.4236/jamp.2019.78118