A Real Representation Method for Solving Yakubovich-<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.4924pt" id="M1" height="15.8401pt" version="1.1" viewBox="-0.0657574 -11.3477 7.53797 15.8401" width="7.53797pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g50-107" d="M400 606C400 634 383 656 353 656C316 656 294 620 294 593C294 564 317 545 343 545C375 545 400 573 400 606ZM366 351C379 413 381 451 356 451C323 451 251 408 183 341L199 313C223 335 267 365 277 365C285 365 284 354 277 312C245 132 222 27 193 -100C182 -148 160 -188 131 -188C113 -188 90 -178 75 -170C64 -164 55 -168 48 -175C38 -185 24 -203 24 -222S48 -257 71 -257C89 -257 131 -241 186 -192C243 -141 286 -46 310 74L366 351Z"/></g></svg>-Conjugate Quaternion Matrix Equation

Song, Caiqin; Feng, Jun-e; Wang, Xiaodong; Zhao, Jianli

doi:https://doi.org/10.1155/2014/285086

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2014 | Article ID 285086 | https://doi.org/10.1155/2014/285086

A Real Representation Method for Solving Yakubovich--Conjugate Quaternion Matrix Equation

Caiqin Song,^1,2Jun-e Feng,¹Xiaodong Wang,³and Jianli Zhao⁴

Academic Editor: Ngai-Ching Wong

Received19 Oct 2013

Revised12 Dec 2013

Accepted14 Dec 2013

Published12 Jan 2014

Abstract

A new approach is presented for obtaining the solutions to Yakubovich--conjugate quaternion matrix equation based on the real representation of a quaternion matrix. Compared to the existing results, there are no requirements on the coefficient matrix . The closed form solution is established and the equivalent form of solution is given for this Yakubovich--conjugate quaternion matrix equation. Moreover, the existence of solution to complex conjugate matrix equation is also characterized and the solution is derived in an explicit form by means of real representation of a complex matrix. Actually, Yakubovich-conjugate matrix equation over complex field is a special case of Yakubovich--conjugate quaternion matrix equation . Numerical example shows the effectiveness of the proposed results.

1. Introduction

The linear matrix equation , which is called the Kalman-Yakubovich matrix equation in [1], is closely related to many problems in conventional linear control systems theory, such as pole assignment design [2], Luenberger-type observer design [3, 4], and robust fault detection [5, 6]. In recent years, many studies have been reported on the solutions to many algebraic equations including quaternion matrix equations and nonlinear matrix equations. Yuan and Liao [7] investigated the least squares solution of the quaternion -conjugate matrix equation (where denotes the -conjugate of quaternion matrix ) with the least norm using the complex representation of quaternion matrix, the Kronecker product of matrices, and the Moore-Penrose generalized inverse. The authors in [8] considered the matrix nearness problem associated with the quaternion matrix equation by means of the CCD-Q, GSVD-Q, and the projection theorem in the finite dimensional inner product space. In addition, Song et al. [9, 10] established the explicit solutions to the quaternion -conjugate matrix equation , , but here the known quaternion matrix is a block diagonal form. Wang et al. in [11, 12] investigated Hermitian tridiagonal solutions and the minimal-norm solution with the least norm of quaternionic least squares problem in quaternionic quantum theory. Besides, in [13, 14], some solutions for the Kalman-Yakubovich equation are presented in terms of the coefficients of characteristic polynomial of matrix or the Leverrier algorithm. The existence of solution to the matrix equation , which, for convenience, is called the Kalman-Yakubovich-conjugate matrix equation, is established, and the explicit solution is derived. Several necessary and sufficient conditions for the existence of a unique solution to the matrix equation over quaternion field are obtained [15]. The authors in [16–18] have provided the consistence of the matrix equation via the consimilarity of two matrices. In [19], Wu et al. construct some explicit expressions of the solution of the matrix equation by means of a real representation of a complex matrix. It is shown that there exists a unique solution if and only if and have no common eigenvalues.

In this paper, we study quaternion -conjugate matrix equation by means of real representation of a quaternion matrix. Compared to the complex representation method [9, 10], the real representation method does not require any special case of the known matrix . We propose the explicit solutions to the above Yakubovich--conjugate quaternion matrix equation. As the special case of quaternion -conjugate matrix equation , complex conjugate matrix equation and Kalman-Yakubovich quaternion matrix equation are also investigated. The explicit solutions to the complex conjugate matrix equation have been established.

Throughout this paper, we use the following notations. Let denote the real number field, the complex number field, and the quaternion field, where , . or denotes the set of all matrices on . For any matrix , , , , , and represent the transpose, conjugate, conjugate transpose, determinant, and adjoint of , respectively. In addition, symbol is the real representation of quaternion matrix . denotes the Kronecker product of two matrices and . If , let , where , , and define to be the -conjugate of . For , is defined as . Furthermore, letting , , and , we have the following notations associated with these matrices: Obviously, is the controllability matrix of the matrix pair , is the observability matrix of the matrix pair , and is a symmetric matrix.

2. Quaternion--Conjugate Matrix Equation

2.1. Real Matrix Equation

In this subsection, we investigate the Yakubovich matrix equation over real field

Theorem 1. Suppose the real matrices , , , ; let Then, all the solutions to the Yakubovich matrix equation (2) can be established as where the matrix is an arbitrary matrix.

Proof. We first show that the matrices and given in (4) are solutions of the matrix equation (2). By the direct calculation we have Due to the relation , it is easily derived that So one has Thus, the matrices and given in (4) satisfy the matrix equation (2).
Secondly, we show the completeness of solution (4). It follows from Theorem 6 of [20] that there are degrees of freedom in the solution of matrix equation (2), while solution (4) has exactly parameters represented by the elements of the free matrix . Therefore, in the following we only need to show that all the parameters in the matrix contribute to the solution. To do this, it suffices to show that the mapping defined by (5) is injective. This is true since is nonsingular under the condition of . The proof is thus completed.

In [21], we can find the following well-known generalized Faddeev-Leverrier algorithm: where , , are the coefficients of the characteristic polynomial of the matrix , and , , are the coefficient matrices of the adjoint matrix .

Theorem 2. Given matrices , , , let Then the matrices and given by (4) have the following equivalent form:

Proof. According to (8), the following is easily obtained: This relation can be compactly expressed as Substituting this into the expression of in (10) and recording the sum, we have Combining this with Theorem 1 gives the conclusion.

2.2. Real Representation of a Quaternion Matrix

For any quaternion matrix , , the real representation matrix of quaternion matrix can be defined as

For a quaternion matrix , we define . In addition, if we let in which is a identity matrix, then , , , are unitary matrices.

The real representation has the following properties, which are given in [13].

Proposition 3. Let , , . Then(1), , ;(2); (3), , , ;(4)the quaternion matrix is nonsingular if and only if is nonsingular, and the quaternion matrix is an unitary matrix if and only if is an orthogonal matrix;(5)if , then ;(6), , , and is even, then

Proposition 4. If is a characteristic value of , then so are , .

For any , let the characteristic polynomial of the real representation matrix be , and define . So by Propositions 3 and 4 we have the following proposition.

Proposition 5. Let , . Then(1) is a real polynomial, and ;(2) is a real polynomial, and ;(3), in which , are real polynomials.

Proof. By Proposition 4, we easily know that is a real number, and . For any , by Proposition 3, we have , so we can obtain the result (3).

2.3. On Solutions to the Quaternion -Conjugate Matrix Equation

In this subsection, we discuss the solution of the following quaternion matrix equation: by means of real representation, where , , and are known matrices, and are unknown matrices.

We first define the real representation of quaternion matrix equation (17) by

According to (1) in Proposition 3, the quaternion matrix equation (17) is equivalent to the following equation: Therefore, the matrix equation (17) can be converted into Thus, we have the following conclusion.

Proposition 6. Given the quaternion matrices , and , then the quaternion matrix equation (17) has a solution if and only if the real representation matrix equation (18) has a solution .

Theorem 7. Let , , and . Then quaternion matrix equation (17) has a solution if and only if real representation matrix equation (18) has a solution . Furthermore, if is a solution to (18), then the following quaternion matrices are solutions to quaternion matrix equation (17):

Proof. By (3) of Proposition 3, the quaternion matrix equation (18) is equivalent to After multiplying the two sides of quaternion matrix equation (22) by , we can obtain Before multiplying the two sides of quaternion matrix equation (23) by , we have Noting that , , we give This shows that if is a real solution of matrix equation (18), then is also a real solution of quaternion matrix equation (18). In addition, according to (3) of Proposition 3, the quaternion matrix equation (18) is also equivalent to After multiplying the two sides of quaternion matrix equation (26) by , we have Noting that , , before multiplying the two sides of the quaternion matrix equation (27) by , gives This is to say that if is a real solution of matrix equation (18), then is also a real solution of matrix equation (18). Similarly, we can prove that is also a real solution of quaternion matrix equation (18). In this case, the conclusion can be obtained along the line of the proof of Theorem 4.2 in [13].

Theorem 8. Given the quaternion matrices , , and , let Then the matrices , are given by in which is an arbitrary quaternion matrix.

Proof. If Yakubovich quaternion -conjugate matrix equation (17) has solution , then real representation matrix equation (18) has solution with the free parameter . By Theorems 2 and 7, we have In addition, by Proposition 5, is a real polynomial and . So according to Proposition 3, we obtain Thus, the conclusion above has been proved.

In the following, we provide an equivalent statement of Theorem 8.

Theorem 9. Given quaternion matrices , , and , let Then the matrices , given by (30) have the following equivalent form: in which is an arbitrary quaternion matrix.

Proof. By the direct computation, we have Thus, the first conclusion has been proved. With this the second conclusion is obviously true.

Finally, we consider the solution to the so-called Kalman-Yakubovich -conjugate quaternion matrix equation Based on the main result proposed above, we have the following conclusions regarding the matrix equation (36).

Corollary 10. Given quaternion matrices , , and , let If is a solution of equation (36), then

Proof. If is a solution of equation (36), then is a solution of the equation . By Theorem 3 in [22] and Proposition 3, we have By Proposition 5, is a real polynomial and . So from Proposition 3 and (39), we have Thus, the first conclusion has been proved. With this the second conclusion is obviously true.

In the following, we provide an equivalent statement of Theorem 7.

Corollary 11. Given quaternion matrices , , and , let If is a solution of (36), then

3. Complex Conjugate Matrix Equation

In this section, we study the solution to the complex matrix equation where , , and . Next, we define real representation of complex matrix as follows.

For any complex matrix , , we define a real representation of a complex matrix as Then the real matrix is called real representation of complex matrix .

Let in which is identity matrix. Then , are unitary matrices. The real presentation has the following properties, which are given by Jiang and Wei [14].

Proposition 12. Consider the following.(1)If , , then , , ; (2)let , , , then ;(3)if , then complex matrix is nonsingular if and only if is nonsingular;(4)if , then ;(5)if , then .

Actually, since complex matrix is a special case of quaternion matrix, in this case, we also have the following similar results. Because the proofs are similar to Section 2 and are omitted.

Theorem 13. Given complex matrices , , and . Let Then the solution to the matrix equation (43) is given by

In the following, we provide an equivalent statement of Theorem 13.

Theorem 14. Given complex matrices , , and , let Then the matrices and given by (47) have the following equivalent form:

Finally, we consider the solution to the so-called Kalman-Yakubovich-conjugate matrix

Based on the main result proposed above, we have the following conclusions regarding matrix equation (50).

Theorem 15. Given the complex matrices , , and , let (1) If is a solution of (50), then (2) If is the unique solution of (50), then

Theorem 16. Given the complex matrices , , and , let (1) If is a solution of (50), then (2) If is the unique solution of (50), then

4. Illustrative Example

In this section, we give an example to obtain the solution of complex conjugate matrix equation .

Example 1. Consider Yakubovich-conjugate matrix equation in the form of (43) with the following parameters:
According to the definition of real representation of a complex matrix, we have By some simple computations, we have Choose then it follows from Theorem 14 that the solution of (43) is

5. Conclusions

In the present paper, by means of the real representation of a quaternion matrix, we study the quaternion matrix equation . Compared to our previous results [10], there are no requirements on the coefficient matrix . Explicit solutions to this quaternion matrix equation are established by application of the real representation of a quaternion matrix. As a special case of quaternion -conjugate matrix equation, complex conjugate matrix equation is also considered and the explicit solutions to complex conjugate are proposed. In addition, the equivalent forms of the explicit solutions are given.

Conflict of Interests

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

Acknowledgments

The authors are very grateful to the anonymous reviewers and the editor for their helpful comments and suggestions which have helped us in improving the quality of this paper. This project is granted financial support from the NNSF (nos. 61374025, 61174141, 11171226, 11301247) of China, the Postdoctoral Science Foundation of China (no. 2013M541900), the Research Awards Young and Middle-Aged Scientists of Shandong Province (BS2011SF009, BS2011DX019), and the Excellent Youth Foundation of Shandong’s Natural Scientific Committee (JQ201219).

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Copyright © 2014 Caiqin Song 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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