International Journal of Mathematics and Mathematical SciencesVolume 2012 (2012), Article ID 307036, 19 pagesdoi:10.1155/2012/307036

Research Article

The Real and Complex Hermitian Solutions to a System of Quaternion Matrix Equations with Applications

Shao-Wen Yu

Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

Received 9 October 2011; Accepted 17 November 2011

Academic Editor: Qing-Wen Wang

Copyright © 2012 Shao-Wen Yu. 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.

Abstract.

We establish necessary and sufficient conditions for the existence of and the expressions for the general real and complex Hermitian solutions to the classical system of quaternion matrix equations . Moreover, formulas of the maximal and minimal ranks of four real matrices , and in solution to the system mentioned above are derived. As applications, we give necessary and sufficient conditions for the quaternion matrix equations , and to have real and complex Hermitian solutions.

1. Introduction

Throughout this paper, we denote the real number field by ; the complex field by ; the set of all matrices over the quaternion algebra by ; the identity matrix with the appropriate size by ; the transpose, the conjugate transpose, the column right space, the row left space of a matrix over by , , respectively; the dimension of by . By [1], for a quaternion matrix . is called the rank of a quaternion matrix and denoted by . The Moore-Penrose inverse of matrix over by which satisfies four Penrose equations , and . In this case, is unique and . Moreover, and stand for the two projectors , and induced by . Clearly, and are idempotent and satisfies , and .

Hermitian solutions to some matrix equations were investigated by many authors. In 1976, Khatri and Mitra [2] gave necessary and sufficient conditions for the existence of the Hermitian solutions to the matrix equations and over the complex field , and presented explicit expressions for the general Hermitian solutions to them by generalized inverses when the solvability conditions were satisfied. Matrix equation that has symmetric patterns with Hermitian solutions appears in some application areas, such as vibration theory, statistics, and optimal control theory ([3–7]). Groß in [8], and Liu et al. in [9] gave the solvability conditions for Hermitian solution and its expressions of over in terms of generalized inverses, respectively. In [10], Tian and Liu established the solvability conditions for to have a common Hermitian solution over by the ranks of coefficient matrices. In [11], Tian derived the general common Hermitian solution of (1.4). Wang and Wu in [12] gave some necessary and sufficient conditions for the existence of the common Hermitian solution to equations for operators between Hilbert -modules by generalized inverses and range inclusion of matrices.

As is known to us, extremal ranks of some matrix expressions can be used to characterize nonsingularity, rank invariance, range inclusion of the corresponding matrix expressions, as well as solvability conditions of matrix equations ([4, 7, 9–24]). Real matrices and its extremal ranks in solutions to some complex matrix equation have been investigated by Tian and Liu ([9, 13–15]). Tian [13] gave the maximal and minimal ranks of two real matrices and in solution to over with its applications. Liu et al. [9] derived the maximal and minimal ranks of the two real matrices and in a Hermitian solution of (1.3), where . In order to investigate the real and complex solutions to quaternion matrix equations, Wang and his partners have been studying the real matrices in solutions to some quaternion matrix equations such as , recently ([24–27]). To our knowledge, the necessary and sufficient conditions for (1.5) over to have the real and complex Hermitian solutions have not been given so far. Motivated by the work mentioned above, we in this paper investigate the real and complex Hermitian solutions to system (1.5) over and its applications.

This paper is organized as follows. In Section 2, we first derive formulas of extremal ranks of four real matrices , and in quaternion solution to (1.5) over , then give necessary and sufficient conditions for (1.5) over to have real and complex solutions as well as the expressions of the real and complex solutions. As applications, we in Section 3 establish necessary and sufficient conditions for (1.6) over to have real and complex solutions.

2. The Real and Complex Hermitian Solutions to System (1.5) Over

In this section, we first give a solvability condition and an expression of the general Hermitian solution to (1.5) over , then consider the maximal and minimal ranks of four real matrices , and in solution to (1.5) over , last, investigate the real and complex Hermitian solutions to (1.5) over .

For an arbitrary matrix where , and are real matrices, we define a map from to by By (2.1), it is easy to verify that satisfies the following properties.(a). (b).(c), . (d), where , (e). (f), .

The following lemmas provide us with some useful results over , which can be generalized to .

Lemma 2.1 (see [2, Lemma  2.1]). Let be known, unknown; then the system (1.3) has a Hermitian solution if and only if In that case, the general Hermitian solution of (1.3) can be expressed as where is arbitrary matrix over with compatible size.

Lemma 2.2 (see [12, Corollary  3.4]). Let be known, unknown, and ; then the system (1.5) have a Hermitian solution if and only if In that case, the general Hermitian solution of (1.5) can be expressed as where is arbitrary matrix over with compatible size.

Lemma 2.3 (see [21, Lemma  2.4]). Let , and . Then they satisfy the following rank equalities. (a). (b). (c). (d).

Lemma 2.3 plays an important role in simplifying ranks of various block matrices.

Lemma 2.4 (see [11, Theorem  4.1, Corollary  4.2]). Let , and be given; then where If ,

Lemma 2.5 (see [28, Theorem  3.1]). Let , and be given. Then the matrix equation is consistent if and only if

Theorem 2.6. System (1.5) has a Hermitian solution over if and only if the system of matrix equations has a symmetric solution over . In that case, the general Hermitian solution of (1.5) over can be written as where are the general solutions of (2.13) over . Written in an explicit form, , and in (2.14) are where is a particular symmetric solution to (2.13), and is arbitrary real matrices with compatible sizes.

Proof. Suppose that (1.5) has a Hermitian solution over . Applying properties (a) and (b) of to (1.5) yields implying that is a real symmetric solution to (2.13).Conversely, suppose that (2.13) has a real symmetric solution That is, then by property of , Hence, implying that , and are also symmetric solutions of (2.13). Thus, is a symmetric solution of (2.13), where Let Then by (2.1), Hence, by the property (a), we know that is a Hermitian solution of (1.5). Observe that , in (2.13) can be written as From Lemma 2.2, the general Hermitian solution to (2.13) can be written as where is arbitrary. Hence, where , substituting them into (2.14), yields the four real matrices , and in (2.15)–(2.18).

Now we consider the maximal and minimal ranks of four real matrices , and in solution to (1.5) over .

Theorem 2.7. Suppose that system (1.5) over has a Hermitian solution, and , , , Then the maximal and minimal ranks of , in Hermitian solution to (1.5) are given by where

Proof. We only prove the case that . Similarly, we can get the results that . Let note that is Hermtian; then is symmetric; hence (2.15) can be written as Note that and ; applying (2.9) and (2.10) to (2.37) yields Let Note that is a particular solution to (2.13), it is not difficult to find by Lemma 2.3, block Gaussian elimination, and property of that Note that , then ; hence Similarly, we can obtain Substituting (2.41) and (2.43) into (2.38) and (2.39) yields (2.33) and (2.34), that is .

Corollary 2.8. Suppose system (1.5) over have a Hermitian solution. Then we have the following. (a) System (1.5) has a real hermtian solution if and only if hold when . In that case, the real solution of (1.5) can be expressed as in (2.15). (b) System (1.5) has a complex solution if and only if (2.44) hold when or or . In that case, the complex solutions of (1.5) can be expressed as or or , where , and are expressed as (2.15), (2.16), (2.17), and (2.18), respectively.

Proof. From (2.34) we can get the necessary and sufficient conditions for . Thus we can get the results of this Corollary.

3. Solvability Conditions for Real and Complex Hermitian Solutions to (1.6) Over

In this section, using the results of Theorem 2.6, Theorem 2.7, and Corollary 2.8, we give necessary and sufficient conditions for (1.6) over to have real and complex Hermitian solutions.

Theorem 3.1. Let , and be defined in Lemma 2.2, , and suppose that system (1.5) and the matrix equation over have Hermitian solutions and , respectively. Then system (1.6) over has a real Hermitian solution if and only if (2.44) hold when , and where

Proof. From Corollary 2.8, system (1.5) over has a real Hermitian solution if and only if (2.44) hold when . By (2.15), the real Hermitian solutions of (1.5) over can be expressed as where is arbitrary matrices with compatible sizes.Let ; in Corollary 2.8 and (2.15). It is easy to verify that the matrix equation over has a real Hermitian solution if and only if (3.1) hold and the real Hermitian solution can be expressed as where is a particular solution to and is arbitrary matrices with compatible sizes. The expression of can also be obtained from Lemma 2.1. Let Equating and , we obtain the following equation: It is obvious that system (1.5) and the matrix equation over have common real Hermitian solution if and only if . Hence, we have the matrix equation We know by Lemma 2.5 that (3.9) is solvable if and only if the following four rank equalities hold Under the conditions that the system (1.5) and the matrix equation over have Hermitian solutions, it is not difficult to show by Lemma 2.3 and block Gaussian elimination that (3.10) are equivalent to the four rank equalities (3.2) and (3.3), respectively. Note that the processes are too much tedious; we omit them here. Obviously, the system (1.5) and the matrix equation over have a common real Hermitian solution if and only if (3.2) and (3.3) hold. Thus, the system (1.6) over has a real Hermitian solution if and only if (2.44) hold when , and (3.1)–(3.3) hold.Similarly, from Corollary 2.8, we know that the system (1.5) over has a complex Hermitian solution if and only if (2.44) hold when , , or; ; its complex Hermitian solutions can be expressed as , or . It is also easy to derive the necessary and sufficient condition for the matrix equation over to have a complex Hermitian solution; its complex Hermitian solution can be expressed as or By equating and and , and and , respectively, we can derive the necessary and sufficient conditions for the system (1.6) over to have a complex Hermitian solution.

Acknowledgment

This research was supported by the Excellent Young Teacher Grant of East China University of Science and Technology (yk0157124) and the Natural Science Foundation of China (11001079).

References

T. W. Hungerford, Algebra, vol. 73 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1980.

C. G. Khatri and S. K. Mitra, “Hermitian and nonnegative definite solutions of linear matrix equations,” SIAM Journal on Applied Mathematics, vol. 31, no. 4, pp. 579–585, 1976.

D. Hua and P. Lancaster, “Linear matrix equations from an inverse problem of vibration theory,” Linear Algebra and Its Applications, vol. 246, pp. 31–47, 1996.

Y. H. Liu and Y. G. Tian, “More on extremal ranks of the matrix expressions with statistical applications,” Numerical Linear Algebra with Applications, vol. 15, no. 4, pp. 307–325, 2008.

M. Wei and Q. Wang, “On rank-constrained Hermitian nonnegative-definite least squares solutions to the matrix equation ,” International Journal of Computer Mathematics, vol. 84, no. 6, pp. 945–952, 2007.

X. Zhang and M.-Y. Cheng, “The rank-constrained Hermitian nonnegative-definite and positive-definite solutions to the matrix equation ,” Linear Algebra and Its Applications, vol. 370, pp. 163–174, 2003.

Y. H. Liu and Y. G. Tian, “Max-min problems on the ranks and inertias of the matrix expressions with applications,” Journal of Optimization Theory and Applications, vol. 148, no. 3, pp. 593–622, 2011.

J. Groß, “A note on the general Hermitian solution to ,” Bulletin of the Malaysian Mathematical Society. Second Series, vol. 21, no. 2, pp. 57–62, 1998.

Y. H. Liu, Y. G. Tian, and Y. Takane, “Ranks of Hermitian and skew-Hermitian solutions to the matrix equation ,” Linear Algebra and Its Applications, vol. 431, no. 12, pp. 2359–2372, 2009.

Y. G. Tian and Y. H. Liu, “Extremal ranks of some symmetric matrix expressions with applications,” SIAM Journal on Matrix Analysis and Applications, vol. 28, no. 3, pp. 890–905, 2006.

Y. G. Tian, “Maximization and minimization of the rank and inertia of the Hermitian matrix expression with applications,” Linear Algebra and Its Applications, vol. 434, no. 10, pp. 2109–2139, 2011.

Q. W. Wang and Z.-C. Wu, “Common Hermitian solutions to some operator equations on Hilbert -modules,” Linear Algebra and its Applications, vol. 432, no. 12, pp. 3159–3171, 2010.

Y. G. Tian, “Ranks of solutions of the matrix equation ,” Linear and Multilinear Algebra, vol. 51, no. 2, pp. 111–125, 2003.

Y. H. Liu, “Ranks of solutions of the linear matrix equation ,” Computers & Mathematics with Applications, vol. 52, no. 6-7, pp. 861–872, 2006.

Y. H. Liu, “Ranks of least squares solutions of the matrix equation ,” Computers & Mathematics with Applications, vol. 55, no. 6, pp. 1270–1278, 2008.

Y. G. Tian, “Upper and lower bounds for ranks of matrix expressions using generalized inverses,” Linear Algebra and Its Applications, vol. 355, pp. 187–214, 2002.

Y. G. Tian and S. Cheng, “The maximal and minimal ranks of with applications,” New York Journal of Mathematics, vol. 9, pp. 345–362, 2003.

Y. G. Tian, “The maximal and minimal ranks of some expressions of generalized inverses of matrices,” Southeast Asian Bulletin of Mathematics, vol. 25, no. 4, pp. 745–755, 2002.

Y. G. Tian, “The minimal rank of the matrix expression ,” Missouri Journal of Mathematical Sciences, vol. 14, no. 1, pp. 40–48, 2002.

Q. W. Wang, Z.-C. Wu, and C.-Y. Lin, “Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1755–1764, 2006.

Q. W. Wang, G.-J. Song, and C.-Y. Lin, “Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1517–1532, 2007.

Q. W. Wang, S.-W. Yu, and C.-Y. Lin, “Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications,” Applied Mathematics and Computation, vol. 195, no. 2, pp. 733–744, 2008.

S. W. Yu and G. J. Song, “Extreme ranks of Hermitian solution to a pair of matrix equations,” in Proceedings of the 6th International Workshop on Matrix and Operators, pp. 253–255, Chengdu, China, July 2011.

Q. W. Wang and J. Jiang, “Extreme ranks of (skew-)Hermitian solutions to a quaternion matrix equation,” Electronic Journal of Linear Algebra, vol. 20, pp. 552–573, 2010.

Q. W. Wang, S. W. Yu, and W. Xie, “Extreme ranks of real matrices in solution of the quaternion matrix equation with applications,” Algebra Colloquium, vol. 17, no. 2, pp. 345–360, 2010.

Q. W. Wang, S.-W. Yu, and Q. Zhang, “The real solutions to a system of quaternion matrix equations with applications,” Communications in Algebra, vol. 37, no. 6, pp. 2060–2079, 2009.

Q. W. Wang, H.-S. Zhang, and S.-W. Yu, “On solutions to the quaternion matrix equation ,” Electronic Journal of Linear Algebra, vol. 17, pp. 343–358, 2008.

Y. G. Tian, “The solvability of two linear matrix equations,” Linear and Multilinear Algebra, vol. 48, no. 2, pp. 123–147, 2000.