Journal of Applied Mathematics

Volume 2014 (2014), Article ID 515816, 17 pages

http://dx.doi.org/10.1155/2014/515816

Research Article

Iterative Methods to Solve the Generalized Coupled Sylvester-Conjugate Matrix Equations for Obtaining the Centrally Symmetric (Centrally Antisymmetric) Matrix Solutions

Yajun Xie1,2 and Changfeng Ma1,2

¹School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, China

²College of Fujian Jiangxia, Fuzhou 350108, China

Received 5 February 2014; Accepted 13 June 2014; Published 8 July 2014

Academic Editor: Sazzad Hossien Chowdhury

Copyright © 2014 Yajun Xie and Changfeng Ma. 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.

The iterative method is presented for obtaining the centrally symmetric (centrally antisymmetric) matrix pair solutions of the generalized coupled Sylvester-conjugate matrix equations , . On the condition that the coupled matrix equations are consistent, we show that the solution pair can be obtained within finite iterative steps in the absence of round-off error for any initial value given centrally symmetric (centrally antisymmetric) matrix. Moreover, by choosing appropriate initial value, we can get the least Frobenius norm solution for the new generalized coupled Sylvester-conjugate linear matrix equations. Finally, some numerical examples are given to illustrate that the proposed iterative method is quite efficient.

1. Introduction

Many research papers are involved in the system of matrix equation ([1–33]). The following matrix equation

is a special case of coupled Sylvester linear matrix equations

In [34], an iterative algorithm was constructed to solve (1) for skew-symmetric matrix . Navarra et al. studied a representation of the general solution for the matrix equations , [35]. By Moore-Penrose generalized inverse, some necessary and sufficient conditions on the existence of the solution and the expressions for the matrix equation were obtained in ([36]). Deng et al. give the consistent conditions and the general expressions of the Hermitian solutions for (1) [37]. In addition, by extending the well-known Jacobi and Gauss-Seidel iterations for , Ding et al. gained iterative solutions for matrix equation (1) and the generalized Sylvester matrix equation [38]. The closed form solutions to a family of generalized Sylvester matrix equations were given by utilizing the so-called Kronecker matrix polynomials in ([39]). In recent years, Dehghan and Hajarian considered the solution for the generalized coupled Sylvester matrix equations [40] , and presented a modified conjugate gradient method to solve the matrix equations over generalized bisymmetric matrix pair . Liang and Liu proposed a modified conjugate gradient method to solve the following problem [41]:

In the present paper, we conceive efficient algorithm to solve the following generalized coupled Sylvester-conjugate linear matrix equations for centrally symmetric (centrally antisymmetric) matrix pair :

where , ,    are given constant matrices, and are unknown matrices to be solved. When and , the problem (4) becomes the problem studied in [42]. When , , and , this system becomes the Yakubovich-conjugate matrix equation investigated in [43]. When , , and , the problem (4) becomes the equation considered in [44]. When and , the problem (4) becomes the equation in [45]. When , , and , (4) becomes the equation in [46].

It is known that modified conjugate gradient (MCG) method is the most popular iterative method for solving the system of linear equation

where is an unknown vector, is a given matrix, and is constant vector. By the definition of the Kronecker product, matrix equations can be transformed into the system (5). Then the MCG can be applied to various linear matrix equations [44, 45]. Based on this idea, in this paper, we propose a modified conjugate gradient method to solve the system (4) and show that a solution pair can be obtained within finite iterative steps in the absence of round-off error for any initial value given centrally symmetric (centrally antisymmetric) matrix. Furthermore, by choosing appropriate initial value matrix pair, we can obtain the least Frobenius norm solution for (4).

As a matter of convenience, some terminology used throughout the paper follows.

is the set of complex matrices and is the set of all real matrices. For , we write , , , , , , , and to denote the real part, the imaginary part, the conjugation, transpose, conjugate transpose, the inverse, the Frobenius norm, and the column space of the matrix , respectively. denotes the block diagonal matrix, where . For any , , denotes the Kronecker product defined as . For the matrix , denotes the operator defined as . We use to denote the identity matrix of size implied by context.

Definition 1. Let and , where denotes the column vector whose element is 1 and the other elements are zeros. An complex matrix is said to be a centrally symmetric (centrally antisymmetric) matrix if , denote the set of all centrally symmetric (centrally antisymmetric) matrices by .

The rest of this paper is organized as follows. In Section 2, we construct modified conjugate gradient (MCG) method for solving the system (4) and show that a solution pair for (4) can be obtained by the MCG method within finite iterative steps in the absence of round-off error for any initial value given centrally symmetric (centrally antisymmetric) matrix. Furthermore, we demonstrate that the least Frobenius norm solution can be obtained by choosing a special kind of initial matrix. Also we give some numerical examples which illustrate that the introduced iterative algorithm is efficient in Section 3. Conclusions are arranged in Section 4.

2. The Iterative Method for Solving the Matrix Equations (4)

In this section, we present the modified conjugate gradient method (MCG) for solving the system (4). Firstly, we recall that the definition of inner product came from [42].

The inner product in space is defined as

By Theorem 1 in [42], we know that the inner product defined by (6) satisfies the following three axioms:

(1)symmetry: ;(2)linearity in the first argument:

where and are real constants;(3)positive definiteness: , for all .

For all real constants , by (1) and (2), we get

namely, the inner product defined by (6) is linear in the second argument.

By the relation between the matrix trace and the conjugate operation, we get

The norm of a matrix generated by this inner product space is denoted by . Then for , we obtain

What is the relationship between this norm and the Frobenius norm? It is well known that and is a Hermite matrix. Then by the knowledge of algebra, we know that is real; hence, . This shows that . Another interesting relationship is that

That is, .

In the following we present some algorithms. The ordinary conjugate gradient (CG) method to solve (5) is as follows [47].

Algorithm 2 (CG method). Consider the following steps.

Step  1. Input , . Choose the initial vectors and set ; calculate , .

Step  2. If or and , stop; otherwise, calculate

Step  3. Update the sequences

Step  4. Set ; return to Step .

It is known that the size of the linear equation (5) will be large, when (4) is transformed to a linear equation (5) by the Kronecker product. Therefore, the iterative Algorithm 2 will consume much more computer time and memory space once increasing the dimensionality of coefficient matrix.

In view of these considerations, we construct the following so-called modified conjugate gradient (MCG) method to solve (4).

Algorithm 3 (MCG method for centrally symmetric matrix version). Consider the following steps.

Step  1. Input appropriate dimension matrices , , , , and , . Choose the initial matrices and , in Definition 1. Compute

set .

Step  2. If or , , stop; otherwise, go to Step .

Step  3. Update the sequences

where

Step  4. Set ; return to Step .

Algorithm 4 (MCG method for centrally antisymmetric matrix version). Consider the following steps.

Step  1. Input matrices , , , , and , . Choose the initial matrix and , in Definition 1. Compute

set .

Step  2. If or , , stop; otherwise, go to Step .

Step  3. Update the sequences

where

Step  4. Set ; return to Step .

Now, we will show that the sequence matrix pair generated by Algorithm 3 converges to the solution for (4) within finite iterative steps in the absence of round-off error for any initial value over centrally symmetric (centrally antisymmetric) matrix.

Lemma 5. Let the sequences , , , , , and generated by Algorithm 3; then have

Proof. By Algorithm 3 and (20), we get

In a similar way, we can get

This together with the definition of inner product yields that

Then by the updated formulas of , , and , we obtain

which completes the proof.

Lemma 6. Let the sequences, , , and , be generated by Algorithm 3; one has

Proof. Firstly, we prove

By mathematical induction, for , by Lemma 5, and noticing , generated by Algorithm 3, we get

where the second equality is from the fact

In addition, by (19), (20), and Lemma 5, we have

where the second equality is from (6) and the fact

Therefore, (27) holds for .

Suppose that (27) holds, for . For , it follows from Lemma 5, (9) that

where the fourth equality holds by the induction assumption. Combining (19) and (20) and by induction with the above result, we obtain

where the third equality is from Lemma 5.

For , by Lemma 5 and the induction, we have

Analogously, for , then we obtain

In addition, from Lemma 5 and the induction, for , we get

So (27) holds, for . By induction principle, (27) holds, for all . For , we obtain

which completes the proof.

Lemma 7. Suppose that the system of matrix equations (4) is consistent; let be an arbitrary solution pair of (4). Then for any initial matrices , , the sequences , , , , and generated by Algorithm 3 satisfy

Proof. The conclusion is accomplished by mathematical induction.

Firstly, we notice that the sequences pair , generated by Algorithm 3 are all central symmetric matrices since initial matrix pair is centrally symmetric matrix. Then for , it follows from Algorithm 3 that

In the same way, we can get

This shows that

That is, (38) holds, for .

Assume (38) holds, for . For , it follows from the updated formulas of , that

Then

On the other hand, we have

Therefore, by (20) we get

Hence, the proof is completed.

Remark 8. Lemma 7 implies that if (4) is consistent, then when . Conversely, if there exists a positive integer such that and in the iteration process of Algorithm 3, then (4) is inconsistent; we will study this condition with other papers in the future.

Remark 9. The above lemmas are achieved under the assumption that initial value is centrally symmetric matrix. Similarly, if the initial matrix is centrally antisymmetric matrix, we can get the same conclusions easily (see Definition 1). Hence, we need not show these results in detail; in the following content, we only discuss the version when .

Theorem 10. Suppose the system (4) is consistent; then, for any initial matrix , , an exact solution of (4) can be derived at most iteration steps by Algorithm 3.

Proof. Assume , for . It follows from Lemma 7 that for . Then can be derived by Algorithm 3. According to Lemma 6, we know , for , . Then the matrix sequence of is an orthogonal basis of the linear space

where . Since and , for , hence , which completes the proof.

Although the proof is trivial, the consequences of this result are of major importance.

When (4) is consistent, the solution of (4) is not unique. Then we need to find the unique least Frobenius norm solution of (4). Next, we introduce the following lemma.

Lemma 11. Suppose , , and the linear matrix equation has a solution ; then is the unique least Frobenius norm solution of .

For a rigorous proof of this lemma above the reader is referred to [46, 48].

Lemma 12. Suppose , , and the linear matrix equation has a solution . If , then is the unique least Frobenius norm solution of .

Proof. Let and . Then can be written as

It shows that

or

Since

this together with Lemma 11 completes the result.

In order to get the least Frobenius norm solution of (4), we need to transform the problem (4).

Let , . Then the problem (4) can be equivalently written as

or

This together with the definition of the Kronecker product yields

where

By some simple calculating, we have

where

Particularly, when , can be substituted with in (51); then we obtain

where , , (see Definition 1).

Obviously, if , then , where is a matrix. So, from the above analysis, we can get the result.

Lemma 13. Let , , and the linear matrix equation has a solution , where . If , then is the unique least Frobenius norm solution of in (59).

Theorem 14. Suppose the system (4) is consistent; if one chooses the initial matrix pair

where are two arbitrary matrices, especially, taking , the solution given by Algorithm 3 is the unique least Frobenius norm solution of (4).

Proof. If has the form of (60), (61), respectively, by of Algorithm 3, we have

where and . Since

we have

where and . By parity of reasoning, we can prove that

where . This together with Theorem 10 yields that

where . Since

we have

Let , , , and in (59); using Kronecker product, we get

In the same way, we can prove that

This shows that

Notice that and in (59). By (56) and (57), we have

It follows from Lemma 11, Lemma 12, and (59) that

where

for all solutions of (4). Since

we obtain

for all solutions of (4). Hence is the unique least Frobenius norm solution of (4).

3. Numerical Experiments

In this section, we report some numerical results to support our Algorithm 3. The iterations have been carried out by MATLAB R2011b (7.13), Intel(R) Core(TM) i7-2670QM, CPU 2.20 GHZ, RAM 8.GB PC Environment.

Example 1. We consider (4) with the following matrices:

We can show that the matrix equation (4) is consistent. Choose the initial matrix pair , ; by Algorithm 3, we obtain its exact solution:

The corresponding residual and least Frobenius norm are

The results above are presented in Figure 1, where , .

Figure 1: The relative error of solution and the residual for Example 1.

Example 2. Consider (4) with the following matrices:

We can show that the matrix equation (4) is also consistent. Choose the initial matrix pair , ; from Algorithm 3, we obtain its exact solution only the 29th iterative step:

The corresponding residual and least Frobenius norm are

The above results are presented in Figure 2, where , .

Figure 2: The relative error of solution and the residual for Example 2.

4. Conclusion

Iterative method is proposed to solve the generalized coupled Sylvester-conjugate linear matrix equations , for center-symmetry (center-antisymmetry) matrix pair . When (4) is consistent, utilizing the Kronecker product, it has been revealed that an exact solution can be obtained by the proposed algorithm within finite iterative steps in the absence of round-off error for any initial value chosen center-symmetry (center-antisymmetry) matrix. Furthermore, we show that the least Frobenius norm solution is obtained by choosing a special kind of initial matrix. Finally, some numerical examples were given to show the efficiency for the proposed method.

Conflict of Interests

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

Acknowledgments

The project was supported by National Natural Science Foundation of China (Grant no. 11071041), Fujian Natural Science Foundation (Grant no. 2013J01006), and the university special fund project of Fujian (Grant no. JK2013060).

References

A. Andrew, “Eigenvectors of certain matrices,” Linear Algebra and Its Applications, vol. 7, pp. 151–162, 1973.

Z. Bai, “The inverse eigenproblem of centrosymmetric matrices with a submatrix constraint and its approximation,” SIAM Journal on Matrix Analysis and Applications, vol. 26, no. 4, pp. 1100–1114, 2005.

J. K. Baksalary and R. Kala, “The matrix equation ,” Linear Algebra and Its Applications, vol. 30, pp. 141–147, 1980.

W. Chen, X. Wang, and T. Zhong, “The structure of weighting coefficient matrices of harmonic differential quadrature and its applications,” Communications in Numerical Methods in Engineering, vol. 12, no. 8, pp. 455–459, 1996.

P. G. Ciarlet, Introduction to Numerical Linear Algebra and Optimisation, Cambridge University Press, Cambridge, Mass, USA, 1989.

L. Datta and S. Morgera, “Some results on matrix symmetries and a pattern recognition application,” IEEE Transactions on Signal Processing, vol. 34, pp. 992–994, 1986.

L. Datta and S. D. Morgera, “On the reducibility of centrosymmetric matrices—applications in engineering problems,” Circuits, Systems, and Signal Processing, vol. 8, no. 1, pp. 71–96, 1989.

J. Delmas, “On adaptive EVD asymptotic distribution of centre-symmetric covariance matrices,” IEEE Transactions on Signal Processing, vol. 47, no. 5, pp. 1402–1406, 1999.

Y. Deng, Z. Bai, and Y. Gao, “Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations,” Numerical Linear Algebra with Applications, vol. 13, no. 10, pp. 801–823, 2006.

G. Konghua, X. Y. Hu, and L. Zhang, “A new iteration method for the matrix equation ,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 1434–1441, 2007.

H. Dai, “On the symmetric solutions of linear matrix equations,” Linear Algebra and its Applications, vol. 131, pp. 1–7, 1990.

S. Karimi and F. Toutounian, “The block least squares method for solving nonsymmetric linear systems with multiple right-hand sides,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 852–862, 2006.

J. F. Li, Z. Y. Peng, and J. J. Peng, “The bisymmetric solution of matrix equation over a matrix inequality constraint,” Mathematica Numerica Sinica, vol. 35, no. 2, pp. 137–150, 2013 (Chinese).

J. R. Li and J. White, “Low rank solution of Lyapunov equations,” SIAM Journal on Matrix Analysis and Applications, vol. 24, no. 1, pp. 260–280, 2002.

F. Li, L. Gong, X. Hu, and L. Zhang, “Successive projection iterative method for solving matrix equation ,” Journal of Computational and Applied Mathematics, vol. 234, no. 8, pp. 2405–2410, 2010.

P. G. Martinsson, V. Rokhlin, and M. Tygert, “A fast algorithm for the inversion of general Toeplitz matrices,” Computers & Mathematics with Applications, vol. 50, no. 5-6, pp. 741–752, 2005.

J. J. Moreau, “Decomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires,” Comptes Rendus de l'Académie des Sciences de Paris A, vol. 225, pp. 238–240, 1962.

T. Penzl, “A cyclic low-rank Smith method for large sparse Lyapunov equations,” SIAM Journal on Scientific Computing, vol. 21, no. 4, pp. 1401–1418, 2000.

Z. Peng, L. Wang, and J. Peng, “The solutions of matrix equation over a matrix inequality constraint,” SIAM Journal on Matrix Analysis and Applications, vol. 33, no. 2, pp. 554–568, 2012.

N. Shinozaki and M. Sibuya, “Consistency of a pair of matrix equations with an application,” Keio Science and Technology Reports, vol. 27, no. 10, pp. 141–146, 1974.

W. F. Trench, “Inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry,” Linear Algebra and Its Applications, vol. 380, pp. 199–211, 2004.

F. Toutounian and S. Karimi, “Global least squares method (Gl-LSQR) for solving general linear systems with several right-hand sides,” Applied Mathematics and Computation, vol. 178, no. 2, pp. 452–460, 2006.

W. F. Trench, “Minimization problems for -symmetric and -skew symmetric matrices,” Linear Algebra and its Applications, vol. 389, pp. 23–31, 2004.

Q.-W. Wang, “Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations,” Computers & Mathematics with Applications, vol. 49, no. 5-6, pp. 641–650, 2005.

Q. Wang, J. Sun, and S. Li, “Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra,” Linear Algebra and Its Applications, vol. 353, pp. 169–182, 2002.

Q. W. Wang and F. Zhang, “The reflexive re-nonnegative definite solution to a quaternion matrix equation,” Electronic Journal of Linear Algebra, vol. 17, pp. 88–101, 2008.

Q. W. Wang, H. X. Chang, and Q. Ning, “The common solution to six quaternion matrix equations with applications,” Applied Mathematics and Computation, vol. 198, no. 1, pp. 209–226, 2008.

Q. Wang and C. Li, “Ranks and the least-norm of the general solution to a system of quaternion matrix equations,” Linear Algebra and its Applications, vol. 430, no. 5-6, pp. 1626–1640, 2009.

Q.-W. Wang, J.-H. Sun, and S.-Z. Li, “Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra,” Linear Algebra and Its Applications, vol. 353, pp. 169–182, 2002.

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.

J. R. Weaver, “Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors,” The American Mathematical Monthly, vol. 92, no. 10, pp. 711–717, 1985.

L. J. Zhao, X. Y. Hu, and L. Zhang, “Least squares solutions to for bisymmetric matrices under a central principal submatrix constraint and the optimal approximation,” Linear Algebra and Its Applications, vol. 428, no. 4, pp. 871–880, 2008.

B. Zhou, G. Duan, and Z. Li, “Gradient based iterative algorithm for solving coupled matrix equations,” Systems & Control Letters, vol. 58, no. 5, pp. 327–333, 2009.

G. Huang, F. Yin, and K. Guo, “An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation ,” Journal of Computational and Applied Mathematics, vol. 212, no. 2, pp. 231–244, 2008.

A. Navarra, P. L. Odell, and D. M. Young, “A representation of the general common solution to the matrix equations A₁XB₁ = C₁ and A₂XB₂ = C₂ with applications,” Computers & Mathematics with Applications, vol. 41, no. 7-8, pp. 929–935, 2001.

F. Piao, Q. Zhang, and Z. Wang, “The solution to matrix equation ,” Journal of the Franklin Institute. Engineering and Applied Mathematics, vol. 344, no. 8, pp. 1056–1062, 2007.

Y. B. Deng, Z. Z. Bai, and Y. H. Gao, “Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations,” Numerical Linear Algebra with Applications, vol. 13, no. 10, pp. 801–823, 2006.

F. Ding, P. X. Liu, and J. Ding, “Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 41–50, 2008.

B. Zhou, Z. Li, G. Duan, and Y. Wang, “Solutions to a family of matrix equations by using the Kronecker matrix polynomials,” Applied Mathematics and Computation, vol. 212, no. 2, pp. 327–336, 2009.

M. Dehghan and M. Hajarian, “An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices,” Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, vol. 34, no. 3, pp. 639–654, 2010.

K. Liang and J. Liu, “Iterative algorithms for the minimum-norm solution and the least-squares solution of the linear matrix equations ,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3166–3175, 2011.

A. G. Wu, G. R. Duan, Y. M. Fu, and W. J. Wu, “Finite iterative algorithms for the generalized Sylvester-conjugate matrix equation ,” Computing, vol. 89, no. 3-4, pp. 147–170, 2010.

A. Wu, G. Feng, J. Hu, and G. Duan, “Closed-form solutions to the nonhomogeneous Yakubovich-conjugate matrix equation,” Applied Mathematics and Computation, vol. 214, no. 2, pp. 442–450, 2009.

T. Jiang and M. Wei, “On solutions of the matrix equations and ,” Linear Algebra and Its Applications, vol. 367, pp. 225–233, 2003.

A. Wu, Y. Fu, and G. Duan, “On solutions of matrix equations and ,” Mathematical and Computer Modelling, vol. 47, no. 11-12, pp. 1181–1197, 2008.

J.-H. Bevis, F.-J. Hall, and R.-E. Hartwing, “Consimilarity and the matrix equation AX − XB = C,” in Current Trends in Matrix Theory, pp. 51–64, North-Holland, New York, NY, USA, 1986.

K. Y Zhang and Z. Xu, Numerical Algebra, Science Press, 2006, (Chinese).

Y. X. Peng, X. Y. Hu, and L. Zhang, “An iterative method for symmetric solutions and optimal approximation solution of the system of matrix equations ,” Applied Mathematics and Computation, vol. 183, no. 2, pp. 1127–1137, 2006.