Algebraic Techniques for Canonical Forms and Applications in Split Quaternionic Mechanics

Jiang, Tongsong; Zhang, Dong; Guo, Zhenwei; Wang, Gang; Vasil’ev, V. I.

doi:https://doi.org/10.1155/2023/4599585

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Abstract Introduction Applications Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 4599585 | https://doi.org/10.1155/2023/4599585

Algebraic Techniques for Canonical Forms and Applications in Split Quaternionic Mechanics

Tongsong Jiang,^1,2,3Dong Zhang,²Zhenwei Guo,²Gang Wang,²and V. I. Vasil’ev²

Academic Editor: Zafar Ullah

Received03 Mar 2023

Revised04 Oct 2023

Accepted12 Oct 2023

Published07 Nov 2023

Abstract

The algebra of split quaternions is a recently increasing topic in the study of theory and numerical computation in split quaternionic mechanics. This paper, by means of a real representation of a split quaternion matrix, studies the problem of canonical forms of a split quaternion matrix and derives algebraic techniques for finding the canonical forms of a split quaternion matrix. This paper also gives two applications for the right eigenvalue and diagonalization in split quaternionic mechanics.

1. Introduction

A split quaternion (or coquaternion), which was found in 1849 by James Cockle, is in the form ofwhere are real numbers. One can easily get that . A quaternion, which was found in 1843 by William Rowan Hamilton, is in the form of , , where are real numbers, and . Let and H denote, respectively, the split quaternion ring and the quaternion ring. The ring and H are two different associative and noncommutative Clifford algebras, and the former ring, which contains zero-divisors, nilpotent elements, and nontrivial idempotents, is a noncommutative ring [1–7].

The split quaternions are used to express Lorentzian rotations, and works on the geometric and mechanical meaning of split quaternions can be found in [8–17]. Moreover, in the study of the relation between complexified classical and non-Hermitian quantum mechanics, there are findings that complexified mechanical systems can alternatively be thought of as certain quaternionic and split quaternionic extensions of the underlying real mechanical systems [12, 18–24]. This identification gives rise to the possibility of using algebraic techniques of split quaternions for dealing with problems in complexified classical and quantum mechanics.

It is known that the theory and algebraic technique for Jordan forms of a matrix play essential roles in the study of matrix theory and its applications. In [25, 26], the authors studied the problems of Jordan forms of complex and quaternion matrices, respectively; for instance, for any quaternion matrices , are uniquely similar to the Jordan form [26], and the authors gave many applications not only in the theory study but also in the numerical computation. This paper, by means of a real representation of a split quaternion matrix, studies the problems of canonical forms of a split quaternion matrix and derives algebraic techniques for finding the canonical forms of a split quaternion matrix. This paper also gives two applications for the right eigenvalue and diagonalization of a split quaternion matrix in split quaternionic mechanics.

Denote the real number field and complex number field, respectively, by R and C and the split quaternion ring by . A split quaternion is called to be a right (left) eigenvalue of a split quaternion matrix if for nonzero vector , and is called to be an eigenvector corresponding to the eigenvalue . For , is said to be nonsingular if for a matrix . Two matrices are said to be similar if for a nonsingular matrix . If is similar to , write , and if is permutation similar to , write . A matrix is called to be diagonalizable if is a diagonal matrix for a nonsingular matrix .

2. Real Representations of a Split Quaternion

Given , , a real representation of was given as [13]and the mapping is an isomorphism of ring onto ring .

For , , , we have the following results:

Therefore, the statements above imply that is isomorphic to , that is, . Therefore, if and only if for two matrices . Moreover, is nonsingular if and only if is nonsingular with .

Remark 1. For , , the real representation of is not unique. The real representation of can also be of the formSimilarly, this real representation also gives an isomorphism of ring onto ring .

Proposition 2 (see [26], chapter 6.7). Let . Then the complex eigenvalues of the real representation given in (2) appear in conjugate pairs, and is similar to the real block-diagonal matrix, that is,where , , are imaginary eigenvalues and are real eigenvalues of real representation , and . Moreover, and are the matrices of and , respectively, as follows.and , is the Jordan block of the eigenvalue .

3. A Canonical Form of a Split Quaternion Matrix

Given , let all different complex eigenvalues of the real representation be as follows:in which are imaginary, , and are real; by Proposition 2, there exists a nonsingular matrix such that is a real block-diagonal matrix in (5), in which and are given in (6), and .

For the cases that the real representation has imaginary eigenvalues, let all matrices in (6) be as follows:

Clearly, for the Jordan blocks , by definition (2), we have

For the case that the real representation has real eigenvalues, let all the Jordan blocks in (10) be as follows:

Suppose that an integer for some is even; then we have

Clearly, by (2), we have

Suppose that an integer for some is odd; since , there exists a Jordan block such that is also odd; set

There are three cases for two odds and as follows: Case 1. . In this case, we have Case 2. . In this case, construct the following split quaternion matrix: and it is clear to get the following equality: Case 3. . In this case, construct the split quaternion matrix:where , .

By the fact that , we easily get the following equality:

It is easy to get by direct calculation

Therefore,

From the aforementioned discussions, we can obtain the following results:

Theorem 3. Let . Then is similar to the following canonical form:in which the split quaternion matrices , , and are defined in (11), (15), and (17), respectively.

Similar to Theorem 3, we can obtain Corollary 4.

Corollary 4. Let . Then is similar to the upper bidiagonal matrix, that is,in which are right split quaternion eigenvalues of the split quaternion matrix , and .

Remark 5. For , , the canonical form of is not unique. For example, if a real representation of is defined in (4), then it is easy to get the following result:By Remark 5, similar to Theorem 3, we can obtain Theorem 6.

Theorem 6. Let , andin which are imaginary eigenvalues, are real eigenvalues of real representation , and and are given in (6). Then is similar to the following canonical form:in which the split quaternion matrices , , and are defined in (27)–(29), respectively.

Corollary 7. Let . Then is similar to the upper bidiagonal matrix, that is,in which are right split quaternion eigenvalues of , and .

From the statements above, we know that the proofs of Theorems 3 and 6 are constructive; they give two algebraic techniques for finding canonical forms by means of real representation of a split quaternion matrix.

4. Applications

This section, by means of canonical forms of a split quaternion matrix, gives two applications for the eigenvalue and diagonalization of a split quaternion matrix.

If for a nonsingular split quaternion , the two split quaternions and are called similar, which can be written as . Denote by the equivalence class that contains . The split quaternion is called to be a principal element of .

By Theorem 3, we get the following result.

Theorem 8. Let , all different complex eigenvalues of the real representation be given in (7). Then there exist four possible different kinds of principal eigenvalues of as follows:in which , , , and are real eigenvalues of and are imaginary eigenvalues of with real .

Moreover, all possible right eigenvalues of are the following equivalence classes with principal element and , respectively.

Remark 9. For , Theorem 8 not only derives all possible right eigenvalues of but also suggests a technique for finding the eigenvalues by means of real representation of .
By the fact that is isomorphic to , for , we know that if and only if . In addition, if is diagonalizable, then by Theorem 3, we get the following result:and by (2) and (3), we haveand the characteristic polynomial of iswhere , , and are real numbers. Therefore, is diagonalizable if and only if satisfies (34) and (35).
From the aforementioned discussions, we can obtain the following results.

Theorem 10. Let . Then is diagonalizable if and only if the characteristic polynomial of has the following form:where , , and are real numbers, and for any eigenvalue of ,where is the multiplicity of the eigenvalue in which case the diagonal matrix is as follows:

Remark 11. Theorem 10 derives an algebraic technique for finding the diagonal split quaternion matrix in (33) by means of real representation of a split quaternion matrix . Moreover, if is diagonalizable, then can be diagonalizable by statement (34).

Example 1. LetBy (2), we haveand by direct calculation, we getand the eigenvalues of are as follows:(1)By Theorem 8, the principal eigenvalues of the split quaternion matrix are and all right eigenvalues of are for any nonsingular split quaternion , .(2)For eigenvalues of real representation matrix , it is easy to get that for , and for , in which is the multiplicity of eigenvalue . Then by Theorem 10, is diagonalizable, and by (38), is similar to diagonal matrix as follows:(3)By (5)–(14), is similar to the following canonical form:and therefore, by Theorem 3, has a canonical form as follows:

5. Conclusions

This paper, by means of the real representation of a split quaternion matrix, studies the problems of canonical forms of a split quaternion matrix and derives algebraic techniques for finding the canonical forms of a split quaternion matrix. This paper also gives two applications for the right eigenvalue and diagonalization of a split quaternion matrix in split quaternionic mechanics. This paper enriches the split quaternion matrix algebraic properties of split quaternionic mechanics and will certainly promote the development of split quaternionic mechanics.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Tongsong Jiang was supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-02-2023-947, February 16, 2023). V. I. Vasil'ev was supported by the Ministry of Science and Higher Education of the Russian Federation (grant no. FSRG-2023-0025). Dong Zhang, Zhenwei Guo, and Gang Wang were supported by the Russian Science Foundation (grant no. 23-71-30013) and Chinese Government Scholarship (CSC nos. 202108370086, 202108370087, and 202008370340).

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Copyright © 2023 Tongsong Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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