An Algebraic Relation between Consimilarity and Similarity of Quaternion Matrices and Applications

Jiang, Tongsong; Cheng, Xuehan; Ling, Sitao

doi:https://doi.org/10.1155/2014/795203

Journal of Applied Mathematics

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Abstract Introduction Preliminaries Applications Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2014 | Article ID 795203 | https://doi.org/10.1155/2014/795203

An Algebraic Relation between Consimilarity and Similarity of Quaternion Matrices and Applications

Tongsong Jiang,^1,2Xuehan Cheng,³and Sitao Ling⁴

Academic Editor: Fan Min

Received12 May 2014

Revised15 Aug 2014

Accepted21 Aug 2014

Published01 Sept 2014

Abstract

This paper, by means of complex representation of a quaternion matrix, discusses the consimilarity of quaternion matrices, and obtains a relation between consimilarity and similarity of quaternion matrices. It sets up an algebraic bridge between consimilarity and similarity, and turns the theory of consimilarity of quaternion matrices into that of ordinary similarity of complex matrices. This paper also gives algebraic methods for finding coneigenvalues and coneigenvectors of quaternion matrices by means of complex representation of a quaternion matrix.

1. Introduction

An antilinear operator is a mapping from one complex vector space into another , which is additive and conjugate homogeneous, that is, for all and any complex number , , in which is the conjugate of . Two complex matrices are said to be consimilar if for some nonsingular complex matrix . Consimilarity of complex matrices arises as a result of studying an antilinear operator referred to different bases in complex vector spaces, and the theory of consimilarity of complex matrices plays an important role in quantum mechanics [1, 2]. Complex consimilarity is an equivalent relation and has been studied [2–5].

In recent years, applications of quaternion matrices are getting more and more important and extensive in quantum mechanics, rigid mechanics, and control theory [6–17]; it is becoming more and more necessary to study the theory and methods of quaternion matrices. In paper [18], the author introduced concepts of consimilarity of quaternion matrices. If and are both quaternion matrices of , they are said to be consimilar if holds for some nonsingular quaternion matrix . Write if is similar to , if is consimilar to , and if is permutation similar to . Permutation similarity is both a similarity and consimilarity relations. A quaternion is said to be the right coneigenvalue attributed to if the con-eigenequation holds. The consimilarity of quaternion matrices is natural extension of that of complex matrices and will have potential applications in the study of theory and numerical computations in modern quantum mechanics, and so forth.

Let be the real number field, the complex number field, and the quaternion field, where , . denotes the set of all matrices on a field . For , let be the transpose of and the conjugate matrix of . For a quaternion with , we use and to denote the conjugate and -conjugate of , respectively. Let , where ; define to be -conjugate of . It is easy to verify that , , and for any two matrices and a matrix with quaternion entries. Obviously, if is a complex matrix.

By means of real representation of complex matrices in paper [5], we studied the properties of consimilarity of complex matrices and gave a relation between consimilarity and similarity of complex matrices. This paper, by means of a complex representation of a quaternion matrix, studies the relation between consimilarity and similarity of quaternion matrices and derives an algebraic relation between consimilarity and similarity on quaternion field. Finally this paper gives an application on coneigenvalues and coneigenvectors of quaternion matrices.

2. Preliminaries

For a quaternion matrix of dimension , denote by , where , . The complex representation of the quaternion matrix is defined [19] to be the complex matrix is known as complex representation of the quaternion matrix . It is easy to verify that .

Let , , . Then by the definition of complex representation we easily get the following results: in which with being the identity matrix, and , . Clearly, from (5) we know that a quaternion matrix is nonsingular if and only if complex representation matrix is nonsingular.

For and , if , then by (4) we have This means that the eigenvalues of complex representation appear in conjugate pairs.

In the same manner, for a Jordan block of an eigenvalue , if , then by (4) we have And this means Jordan blocks of complex representation appear in conjugate pairs.

From the statement above we have the following result.

Proposition 1. Let . Then(1)the real eigenvalues of complex representation appear in pairs, and the imaginary eigenvalues of complex representation appear in conjugate pairs; that is, if complex eigenvalue is an eigenvalue of , then complex eigenvalue is also an eigenvalue of ,(2)the real Jordan blocks of complex representation appear in pairs, and the imaginary Jordan blocks of complex representation appear in conjugate pairs; that is, if is a Jordan block of complex representation related to complex eigenvalue , then is a Jordan block of complex representation related to complex eigenvalue .

The following Proposition 2 comes from the fact that , and by direct calculation for real numbers and , and real matrices and .

Proposition 2. Let be a quaternion matrix. Then(1); this means that is consimilar to ; that is, ;(2)if is a Jordan block, , then ;(3)if is a complex matrix, , then

Based on the quaternion field , the author in [18] gave the following results.

Proposition 3 (see [18] ). If , then

Proposition 4 (see [18] ). If , then in which are right coneigenvalues of , are real numbers, and . is uniquely determined by up to the order of Jordan blocks , and is said to be the Jordan canonical form of under consimilarity.

3. An Algebraic Relation between Consimilarity and Similarity

This section gives an algebraic relation between consimilarity and similarity of quaternion matrices by means of the complex representation in (2).

Let be two quaternion matrices. If is consimilar to , then there exists a nonsingular quaternion matrix such that , by (5) . This means that if is consimilar to , then is similar to .

Conversely, if is similar to , then and have the same eigenvalues. By Proposition 1, let be all eigenvalues and corresponding Jordan blocks. There exists a complex and full-rank matrix by [20, chapter 6.7] such that , . Then by (4), we have and (11) is equivalent to Therefore there exists a nonsingular complex matrix , such that in which , , are real numbers, and .

Let , , and . Then . From the nonsingular complex matrix , we get is a nonsingular quaternion matrix; by (1) and (5), (13) is equivalent to This means that . Let , are real numbers, and , and . Then by Proposition 2 we have , and we have . Similarly . Therefore .

The statement above implies the following result.

Theorem 5. Let . Then is consimilar to if and only if complex representation matrix is similar to complex representation matrix ; that is, if and only if .

Combining Proposition 3 and Theorem 5, we get the following result.

Corollary 6. Let . Then

The proof of the Theorem 5 is constructive, and the following results come from the proof above.

Corollary 7. Let . Then the following statements are equivalent.(1)The Jordan canonical form of complex matrix under similarity is where , , , , are eigenvalues of .(2)The Jordan canonical form of quaternion matrix under consimilarity is where , , , , are coneigenvalues of .

Corollary 8. Let . Then where are coneigenvalues of , , , , , in which are all eigenvalues of complex representation , and corresponding Jordan blocks in (16). is uniquely determined by up to the order of Jordan blocks , and is said to be the Jordan canonical form of under consimilarity.

Corollary 9. Let . Then has at least a coneigenvalue , . If , , , , , are all eigenvalues of complex representation , then are coneigenvalues of quaternion matrix , in which and for any nonzero quaternion , , , are all coneigenvalues of quaternion matrix .

The following result comes immediately from Theorem 5 and the definition of complex representation in (2).

Corollary 10. Let . Then is consimilar to if and only if is similar to .

4. Algebraic Methods and Applications

This section gives an algebraic method for relation between consimilarity and similarity of quaternion matrices by means of complex representation and obtains an algebraic method for coneigenvalues and coneigenvectors for quaternion matrices as an application.

Let , . Suppose there exists a complex matrix such that . By (4) Let . Then Let where . It is easy to get by direct calculation where From (23) we construct a complex matrix using the identities and Clearly . Therefore by (5) and , (21) is equivalent to

The statement above implies the following result.

Proposition 11. Let , . Then there exists a quaternion matrix such that if and only if there exists a complex matrix such that . In which case, there exists a complex matrix such that ; let Then is a quaternion matrix and .

By Theorem 5 and the statement above we have the following result.

Corollary 12. Let . Then is consimilar to if and only if is similar to ; that is, if and only if . In which case, there exists a nonsingular complex matrix such that ; let Then is a quaternion matrix and , and if is a nonsingular, then .

Corollary 12 gives an algebraic method for the relation between consimilarity and similarity of quaternion matrices by means of complex representation and turns theory of consimilarity of quaternion matrices into that of ordinary similarity of complex representation matrices.

As a special case of Corollary 12, we give an algebraic method for relation between coneigenvalues and coneigenvectors of a quaternion matrix and corresponding eigenvalues and eigenvectors of the complex representation matrix as follows.

Corollary 13. Let , . Then there exists a vector such that if and only if there exists a vector such that . In which case, there exists a vector such that ; let Then .

Now we give algebraic methods for finding coneigenvalues and coneigenvectors of a quaternion matrix by means of complex representation.

For , if is a complex eigenvalue of with , , , then by (11) and (12), , , and in which . By Corollary 13, let Then . This means that is a coneigenvalue of and if , then is a coneigenvector related to coneigenvalue .

By the statement above we get the following result.

Corollary 14. Let . If is a complex eigenvalue of complex representation and is eigenvector related to with , let Then . This means if , then is a coneigenvalue of and is a coneigenvector related to coneigenvalue .

Moreover, let , , and . Then by Proposition 2 and the fact that we have in which and . That is is a coneigenvalue of quaternion matrix , and is a coneigenvector related to .

By Corollary 14 and the statement above we have the following result.

Theorem 15. Let . If is a complex eigenvalue of complex representation and is an eigenvector related to with , let Then and . These mean that and are two coneigenvalues of quaternion matrix , and if , then and are corresponding coneigenvectors related to and , respectively. Moreover are right coneigenvalues of , and is corresponding coneigenvector for any nonzero quaternion .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their sincere thanks to the referees for the careful reading and very helpful comments on the earlier versions of this paper. This paper is supported by the National Natural Science Foundation of China (11301529 and 11301252) and Postdoctoral Science Foundation of China (2013M540472).

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Copyright

Copyright © 2014 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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