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

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 techniques for eigenvalues and eigenvectors in quaternionic and split quaternio-nic mechanics.


Introduction
A quaternion, which was found in 1840 by William Rowan Hamilton [1], is in the form of , , , q q q q ∈ R , and ij ji k = − = , jk kj i = − =, ki ik j = − =. A split quaternion, which was found in 1849 by James Cockle [2], is in the form of 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, quantum 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 quaternion 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 quaternion matrices by means of complex representation and companion 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 quaternion, in [12], by means of complex representation of a split quaternion matrix, 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 extension of Gershgorin theorem.
Because there are amazing relationships between quaternions and split quaternions [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 eigenvectors. Therefore, a new algebraic structure is defined, it is v-quaternion.
A v-quaternion is in the form of 1 2 3 4 , , , q q q q ∈ R , and ij ji k Before start this paper, first introduce the necessary symbols and preliminary knowledge. Let R be the real number field, i = ⊕ C R R the complex number field, and ( ) 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 .

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 1 x px q − = , 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 1 u pu q It is easy to prove the following result by direct calculation.
at this point, then construct a nonsingular v-quaternion x such that in which 1 x = . When 2 0 q ≠ , and this case also implies 0 v > , then construct a nonsingular v-quaternion x such that

Journal of Applied Mathematics and Physics
The following two special cases about quaternion and split quaternion comes from i j k q q q q q = + + + is a quaternion, then construct a nonsingular quaternion x such that q q q > + , then construct a nonsingular split quaternion x such that q q q < + , then construct a nonsingular split quaternion x such that Remark 1 The statement above concludes that the set of v-quaternions is divided into four kinds of different equivalence classes. Moreover, if

Complex Representation of v-Quaternion Matrices
For any v-quaternion matrix ( ) ( )

Journal of Applied Mathematics and Physics
It is easy to prove that the mapping A σ is an isomorphism of ring For any From the statement above we have the following results. 2) The v-quaternion matrix A has at least a complex eigenvalue. Moreover, if λ is a complex eigenvalue of A σ with A σ β βλ = , let  Moreover, if there exists

Eigenvalues and Eigenvectors of v-Quaternion Matrices
then it is easy to get similarly that ( ) 11 12 in which ( ) 11 11 22 By the fact that , construct a v-quaternion α , we can get the following equality by the direct calculation. 11 12 The statements above imply following result. In which case, if in which 0

Algebraic Techniques for Eigenvalues and Eigenvectors
Then by (4.5) construct the nonzero vector That is α is an eigenvector of the v-quaternion matrix A related to the right eigenvalue λ .
therefore there exists non-negative real number . Then by (4.5) construct the 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 and clear that Therefore there only exists non-negative real number Then by (4.5) construct the 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 Then by (4.5) construct the  2) Compute all eigenvalues of A σ . Let all the eigenvalues of A σ be as follows.

Algorithm and Example Algorithm
then nonsingular vector That is α is an eigenvector 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- The complex representation of the v-quaternion matrix A is as follows

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 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 techniques for eigenvalues and eigenvectors in quaternionic and split quaternionic mechanics.