Let R be an n by n nontrivial real symmetric involution matrix, that is, R = R⁻¹ = R^T ≠ I_n. An n × n complex matrix A is termed R-conjugate if , where denotes the conjugate of A. We give necessary and sufficient conditions for the existence of the Hermitian R-conjugate solution to the system of complex matrix equations AX = C and XB = D and present an expression of the Hermitian R-conjugate solution to this system when the solvability conditions are satisfied. In addition, the solution to an optimal approximation problem is obtained. Furthermore, the least squares Hermitian R-conjugate solution with the least norm to this system mentioned above is considered. The representation of such solution is also derived. Finally, an algorithm and numerical examples are given.

1. Introduction

Throughout, we denote the complex m × n matrix space by ℂ^m×n, the real m × n matrix space by ℝ^m×n, and the set of all matrices in ℝ^m×n with rank r by . The symbols , and ∥A∥ stand for the identity matrix with the appropriate size, the conjugate, the transpose, the conjugate transpose, the Moore-Penrose generalized inverse, and the Frobenius norm of A ∈ ℂ^m×n, respectively. We use V_n to denote the n × n backward matrix having the elements 1 along the southwest diagonal and with the remaining elements being zeros.

Recall that an n × n complex matrix A is centrohermitian if . Centrohermitian matrices and related matrices, such as k-Hermitian matrices, Hermitian Toeplitz matrices, and generalized centrohermitian matrices, appear in digital signal processing and others areas (see, [1–4]). As a generalization of a centrohermitian matrix and related matrices, Trench [5] gave the definition of R-conjugate matrix. A matrix A ∈ ℂ^n×n is R-conjugate if , where R is a nontrivial real symmetric involution matrix, that is, R = R⁻¹ = R^T and R ≠ I_n. At the same time, Trench studied the linear equation Az = w for R-conjugate matrices in [5], where z, w are known column vectors.

Investigating the matrix equation

()

with the unknown matrix X being symmetric, reflexive, Hermitian-generalized Hamiltonian, and repositive definite is a very active research topic [6–14]. As a generalization of (1.1), the classical system of matrix equations

()

has attracted many author’s attention. For instance, [15] gave the necessary and sufficient conditions for the consistency of (1.2), [16, 17] derived an expression for the general solution by using singular value decomposition of a matrix and generalized inverses of matrices, respectively. Moreover, many results have been obtained about the system (1.2) with various constraints, such as bisymmetric, Hermitian, positive semidefinite, reflexive, and generalized reflexive solutions (see, [18–28]). To our knowledge, so far there has been little investigation of the Hermitian R-conjugate solution to (1.2).

Motivated by the work mentioned above, we investigate Hermitian R-conjugate solutions to (1.2). We also consider the optimal approximation problem

()

where E is a given matrix in ℂ^n×n and S_X the set of all Hermitian R-conjugate solutions to (1.2). In many cases the system (1.2) has not Hermitian R-conjugate solution. Hence, we need to further study its least squares solution, which can be described as follows: Let RHℂ^n×n denote the set of all Hermitian R-conjugate matrices in ℂ^n×n:

()

Find

such that

()

In Section 2, we present necessary and sufficient conditions for the existence of the Hermitian R-conjugate solution to (1.2) and give an expression of this solution when the solvability conditions are met. In Section 3, we derive an optimal approximation solution to (1.3). In Section 4, we provide the least squares Hermitian R-conjugate solution to (1.5). In Section 5, we give an algorithm and a numerical example to illustrate our results.

2. R-Conjugate Hermitian Solution to (1.2)

In this section, we establish the solvability conditions and the general expression for the Hermitian R-conjugate solution to (1.2).

We denote Rℂ^n×n and RHℂ^n×n the set of all R-conjugate matrices and Hermitian R-conjugate matrices, respectively, that is,

()

where R is n × n nontrivial real symmetric involution matrix.

Chang et al. in [29] mentioned that for nontrivial symmetric involution matrix R ∈ ℝ^n×n, there exist positive integer r and n × n real orthogonal matrix [P, Q] such that

()

where P ∈ ℝ^n×r, Q ∈ ℝ^n×(n−r). By (2.2),

()

Throughout this paper, we always assume that the nontrivial symmetric involution matrix R is fixed which is given by (2.2) and (2.3). Now, we give a criterion of judging a matrix is R-conjugate Hermitian matrix.

Theorem 2.1. A matrix K ∈ HRℂ^n×n if and only if there exists a symmetric matrix H ∈ ℝ^n×n such that K = ΓHΓ^*, where

()

with P, Q being the same as (2.2).

Proof. If K ∈ HRℂ^n×n, then . By (2.2),

()

which is equivalent to

()

Suppose that

()

Substituting (2.7) into (2.6), we obtain

()

Hence,

are real matrices. If we denote M = iK₁₂, N = −iK₂₁, then by (2.7)

()

Let Γ = [P, iQ], and

()

Then, K can be expressed as ΓHΓ^*, where Γ is unitary matrix and H is a real matrix. By K = K^*

()

we obtain H = H^T.

Conversely, if there exists a symmetric matrix H ∈ ℝ^n×n such that K = ΓHΓ^*, then it follows from (2.3) that

()

that is, K ∈ HRℂ^n×n.

Theorem 2.1 implies that an arbitrary complex Hermitian R-conjugate matrix is equivalent to a real symmetric matrix.

Lemma 2.2. For any matrix A ∈ ℂ^m×n, A = A₁ + iA₂, where

()

Proof. For any matrix A ∈ ℂ^m×n, it is obvious that A = A₁ + iA₂, where A₁, A₂ are defined as (2.13). Now, we prove that the decomposition A = A₁ + iA₂ is unique. If there exist B₁, B₂ such that A = B₁ + iB₂, then

()

It follows from A₁, A₂, B₁, and B₂ are real matrix that

()

Hence, A = A₁ + iA₂ holds, where A₁, A₂ are defined as (2.13).

By Theorem 2.1, for X ∈ HRℂ^n×n, we may assume that

()

where Γ is defined as (2.4) and Y ∈ ℝ^n×n is a symmetric matrix.

Suppose that AΓ = A₁ + iA₂ ∈ ℂ^m×n, CΓ = C₁ + iC₂ ∈ ℂ^m×n, Γ^*B = B₁ + iB₂ ∈ ℂ^n×l, and Γ^*D = D₁ + iD₂ ∈ ℂ^n×l, where

()

Then, system (1.2) can be reduced into

()

which implies that

()

Let

()

Then, system (1.2) has a solution X in HRℂ^n×n if and only if the real system

()

has a symmetric solution Y in ℝ^n×n.

Lemma 2.3 (Theorem 1 in [7]). Let A ∈ ℝ^m×n. The SVD of matrix A is as follows

()

where U = [U₁, U₂] ∈ ℝ^m×m and V = [V₁, V₂] ∈ ℝ^n×n are orthogonal matrices, Σ = diag (σ₁, …, σ_r), σ_i > 0 (i = 1, …, r), r = rank (A), U₁ ∈ ℝ^m×r, V₁ ∈ ℝ^n×r. Then, (1.1) has a symmetric solution if and only if

()

In that case, it has the general solution

()

where G is an arbitrary (n − r)×(n − r) symmetric matrix.

By Lemma 2.3, we have the following theorem.

Theorem 2.4. Given A ∈ ℂ^m×n, C ∈ ℂ^m×n, B ∈ ℂ^n×l, and D ∈ ℂ^n×l. Let A₁, A₂, C₁, C₂, B₁, B₂, D₁, D₂, F, G, K, L, M, and N be defined in (2.17), (2.20), respectively. Assume that the SVD of M ∈ ℝ^(2m+2l)×n is as follows

()

where U = [U₁, U₂] ∈ ℝ^{(2m+2l)×(2m+2l)} and V = [V₁, V₂] ∈ ℝ^n×n are orthogonal matrices, M₁ = diag (σ₁, …, σ_r), σ_i > 0 (i = 1, …, r), r = rank (M), U₁ ∈ ℝ^(2m+2l)×r, V₁ ∈ ℝ^n×r. Then, system (1.2) has a solution in HRℂ^n×n if and only if

()

In that case, it has the general solution

()

where G is an arbitrary (n − r)×(n − r) symmetric matrix.

3. The Solution of Optimal Approximation Problem (1.3)

When the set S_X of all Hermitian R-conjugate solution to (1.2) is nonempty, it is easy to verify S_X is a closed set. Therefore, the optimal approximation problem (1.3) has a unique solution by [30].

Theorem 3.1. Given A ∈ ℂ^m×n, C ∈ ℂ^m×n, B ∈ ℂ^n×l, D ∈ ℂ^n×l, E ∈ ℂ^n×n, and . Assume S_X is nonempty, then the optimal approximation problem (1.3) has a unique solution and

()

Proof. Since S_X is nonempty, X ∈ S_X has the form of (2.27). By Lemma 2.2, Γ^*EΓ can be written as

()

where

()

According to (3.2) and the unitary invariance of Frobenius norm

()

We get

()

Then,

is consistent if and only if there exists G ∈ ℝ^{(n−r)×(n−r)} such that

()

For the orthogonal matrix V

()

Therefore,

()

is equivalent to

()

Substituting (3.9) into (2.27), we obtain (3.1).

4. The Solution of Problem (1.5)

In this section, we give the explicit expression of the solution to (1.5).

Theorem 4.1. Given A ∈ ℂ^m×n, C ∈ ℂ^m×n, B ∈ ℂ^n×l, and D ∈ ℂ^n×l. Let A₁, A₂, C₁, C₂, B₁, B₂, D₁, D₂, F, G, K, L, M, and N be defined in (2.17), (2.20), respectively. Assume that the SVD of M ∈ ℝ^(2m+2l)×n is as (2.25) and system (1.2) has not a solution in HRℂ^n×n. Then, X ∈ S_L can be expressed as

()

where Y₂₂ ∈ ℝ^{(n−r)×(n−r)} is an arbitrary symmetric matrix.

Proof. It yields from (2.17)–(2.21) and (2.25) that

()

Assume that

()

Then, we have

()

Hence,

()

is solvable if and only if there exist Y₁₁, Y₁₂ such that

()

It follows from (4.6) that

()

Substituting (4.7) into (4.3) and then into (2.16), we can get that the form of elements in S_L is (4.1).

Theorem 4.2. Assume that the notations and conditions are the same as Theorem 4.1. Then,

()

if and only if

()

Proof. In Theorem 4.1, it implies from (4.1) that is equivalent to X has the expression (4.1) with Y₂₂ = 0. Hence, (4.9) holds.

5. An Algorithm and Numerical Example

Base on the main results of this paper, we in this section propose an algorithm for finding the solution of the approximation problem (1.3) and the least squares problem with least norm (1.5). All the tests are performed by MATLAB 6.5 which has a machine precision of around 10⁻¹⁶.

Algorithm 5.1. (1) Input A ∈ ℂ^m×n, C ∈ ℂ^m×n, B ∈ ℂ^n×l, D ∈ ℂ^n×l.

(2) Compute A₁, A₂, C₁, C₂, B₁, B₂, D₁, D₂, F, G, K, L, M, and N by (2.17) and (2.20).

(3) Compute the singular value decomposition of M with the form of (2.25).

(4) If (2.26) holds, then input E ∈ ℂ^n×n and compute the solution of problem (1.3) according (3.1), else compute the solution to problem (1.5) by (4.9).

To show our algorithm is feasible, we give two numerical example. Let an nontrivial symmetric involution be

()

We obtain [P, Q] in (2.2) by using the spectral decomposition of R, then by (2.4)

()

Example 5.2. Suppose A ∈ ℂ^2×4, C ∈ ℂ^2×4, B ∈ ℂ^4×3, D ∈ ℂ^4×3, and

()

We can verify that (2.26) holds. Hence, system (1.2) has an Hermitian R-conjugate solution. Given

()

Applying Algorithm 5.1, we obtain the following:

()

Example 5.2 illustrates that we can solve the optimal approximation problem with Algorithm 5.1 when system (1.2) have Hermitian R-conjugate solutions.

Example 5.3. Let A, B, and C be the same as Example 5.2, and let D in Example 5.2 be changed into

()

We can verify that (2.26) does not hold. By Algorithm 5.1, we get

()

Example 5.3 demonstrates that we can get the least squares solution with Algorithm 5.1 when system (1.2) has not Hermitian R-conjugate solutions.

Acknowledgments

This research was supported by the Grants from the Key Project of Scientific Research Innovation Foundation of Shanghai Municipal Education Commission (13ZZ080), the National Natural Science Foundation of China (11171205), the Natural Science Foundation of Shanghai (11ZR1412500), the Ph.D. Programs Foundation of Ministry of Education of China (20093108110001), the Discipline Project at the corresponding level of Shanghai (A. 13-0101-12-005), Shanghai Leading Academic Discipline Project (J50101), the Natural Science Foundation of Hebei province (A2012403013), and the Natural Science Foundation of Hebei province (A2012205028). The authors are grateful to the anonymous referees for their helpful comments and constructive suggestions.

References

1 Hill R. D., Bates R. G., and Waters S. R., On centro-Hermitian matrices, SIAM Journal on Matrix Analysis and Applications. (1990) 11, no. 1, 128–133, https://doi.org/10.1137/0611009, 1032222.
10.1137/0611009
Web of Science® Google Scholar
2 Kouassi R., Gouton P., and Paindavoine M., Approximation of the Karhunen-Loeve tranformation and Its application to colour images, Signal Processing: Image Communication. (2001) 16, 541–551.
10.1016/S0923-5965(00)00035-7
Web of Science® Google Scholar
3 Lee A., Centro-Hermitian and skew-centro-Hermitian matrices, Linear Algebra and Its Applications. (1980) 29, 205–210, https://doi.org/10.1016/0024%2D3795(80)90241%2D4, 562760.
10.1016/0024-3795(80)90241-4
Google Scholar
4 Liu Z.-Y., Cao H.-D., and Chen H.-J., A note on computing matrix-vector products with generalized centrosymmetric (centrohermitian) matrices, Applied Mathematics and Computation. (2005) 169, no. 2, 1332–1345, https://doi.org/10.1016/j.amc.2004.10.071, 2174724.
10.1016/j.amc.2004.10.071
Web of Science® Google Scholar
5 Trench W. F., Characterization and properties of matrices with generalized symmetry or skew symmetry, Linear Algebra and Its Applications. (2004) 377, 207–218, https://doi.org/10.1016/j.laa.2003.07.013, 2021612, ZBL1046.15028.
10.1016/j.laa.2003.07.013
Web of Science® Google Scholar
6 Chu K.-W. E., Symmetric solutions of linear matrix equations by matrix decompositions, Linear Algebra and Its Applications. (1989) 119, 35–50, https://doi.org/10.1016/0024%2D3795(89)90067%2D0, 1005233, ZBL0688.15003.
10.1016/0024-3795(89)90067-0
Web of Science® Google Scholar
7 Dai H., On the symmetric solutions of linear matrix equations, Linear Algebra and Its Applications. (1990) 131, 1–7, https://doi.org/10.1016/0024%2D3795(90)90370%2DR, 1057060, ZBL0712.15009.
10.1016/0024-3795(90)90370-R
Web of Science® Google Scholar
8 Don F. J. H., On the symmetric solutions of a linear matrix equation, Linear Algebra and Its Applications. (1987) 93, 1–7, https://doi.org/10.1016/S0024%2D3795(87)90308%2D9, 898539, ZBL0622.15001.
10.1016/S0024-3795(87)90308-9
Web of Science® Google Scholar
9 Kyrchei I., Explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations, Linear Algebra and Its Applications. (2013) 438, no. 1, 136–152, https://doi.org/10.1016/j.laa.2012.07.049, 2993371.
10.1016/j.laa.2012.07.049
Web of Science® Google Scholar
10 Li Y., Gao Y., and Guo W., A Hermitian least squares solution of the matrix equation AXB = C subject to inequality restrictions, Computers & Mathematics with Applications. (2012) 64, no. 6, 1752–1760, https://doi.org/10.1016/j.camwa.2012.02.010, 2960799.
10.1016/j.camwa.2012.02.010
Web of Science® Google Scholar
11 Peng Z.-Y. and Hu X.-Y., The reflexive and anti-reflexive solutions of the matrix equation AX = B, Linear Algebra and Its Applications. (2003) 375, 147–155, https://doi.org/10.1016/S0024%2D3795(03)00607%2D4, 2013461, ZBL1050.15016.
10.1016/S0024-3795(03)00607-4
Web of Science® Google Scholar
12 Wu L., The re-positive definite solutions to the matrix inverse problem AX = B, Linear Algebra and Its Applications. (1992) 174, 145–151, 1176457, ZBL0754.15003.
Web of Science® Google Scholar
13 Yuan S. F., Wang Q. W., and Zhang X., Least-squares problem for the quaternion matrix equation AXB + CYD = E over different constrained matrices, International Journal of Computer Mathematics. In presshttps://doi.org/10.1080/00207160.2012.722626, 722626.
10.1080/00207160.2012.722626
Google Scholar
14 Zhang Z.-Z., Hu X.-Y., and Zhang L., On the Hermitian-generalized Hamiltonian solutions of linear matrix equations, SIAM Journal on Matrix Analysis and Applications. (2005) 27, no. 1, 294–303, https://doi.org/10.1137/S0895479801396725, 2176822, ZBL1087.15021.
10.1137/S0895479801396725
CAS Web of Science® Google Scholar
15 Cecioni F., Sopra operazioni algebriche, Annali della Scuola Normale Superiore di Pisa. (1910) 11, 17–20.
Google Scholar
16 Chu K.-W. E., Singular value and generalized singular value decompositions and the solution of linear matrix equations, Linear Algebra and Its Applications. (1987) 88-89, 83–98, https://doi.org/10.1016/0024%2D3795(87)90104%2D2, 882442, ZBL0612.15003.
10.1016/0024-3795(87)90104-2
Web of Science® Google Scholar
17 Mitra S. K., A pair of simultaneous linear matrix equations A₁XB₁ = C₁, A₂XB₂ = C₂ and a matrix programming problem, Linear Algebra and Its Applications. (1990) 131, 107–123, https://doi.org/10.1016/0024%2D3795(90)90377%2DO, 1057067, ZBL0712.15010.
10.1016/0024-3795(90)90377-O
Web of Science® Google Scholar
18 Chang H.-X. and Wang Q.-W., Reflexive solution to a system of matrix equations, Journal of Shanghai University. (2007) 11, no. 4, 355–358, https://doi.org/10.1007/s11741%2D007%2D0407%2D2, 2346769, ZBL1141.15319.
10.1007/s11741-007-0407-2
Google Scholar
19 Dajić A. and Koliha J. J., Equations ax = c and xb = d in rings and rings with involution with applications to Hilbert space operators, Linear Algebra and Its Applications. (2008) 429, no. 7, 1779–1809, https://doi.org/10.1016/j.laa.2008.05.012, 2444361, ZBL1149.47011.
10.1016/j.laa.2008.05.012
Web of Science® Google Scholar
20 Dajić A. and Koliha J. J., Positive solutions to the equations AX = C and XB = D for Hilbert space operators, Journal of Mathematical Analysis and Applications. (2007) 333, no. 2, 567–576, https://doi.org/10.1016/j.jmaa.2006.11.016, 2331678, ZBL1120.47009.
10.1016/j.jmaa.2006.11.016
Web of Science® Google Scholar
21 Dong C.-Z., Wang Q.-W., and Zhang Y.-P., The common positive solution to adjointable operator equations with an application, Journal of Mathematical Analysis and Applications. (2012) 396, no. 2, 670–679, https://doi.org/10.1016/j.jmaa.2012.07.001, 2961260.
10.1016/j.jmaa.2012.07.001
Web of Science® Google Scholar
22 Fang X., Yu J., and Yao H., Solutions to operator equations on Hilbert C^*-modules, Linear Algebra and Its Applications. (2009) 431, no. 11, 2142–2153, https://doi.org/10.1016/j.laa.2009.07.009, 2567821.
10.1016/j.laa.2009.07.009
Web of Science® Google Scholar
23 Khatri C. G. and Mitra S. K., Hermitian and nonnegative definite solutions of linear matrix equations, SIAM Journal on Applied Mathematics. (1976) 31, no. 4, 579–585, 0417212, https://doi.org/10.1137/0131050, ZBL0359.65033.
10.1137/0131050
Web of Science® Google Scholar
24 Kyrchei I., Analogs of Cramer′s rule for the minimum norm least squares solutions of some matrix equations, Applied Mathematics and Computation. (2012) 218, no. 11, 6375–6384, https://doi.org/10.1016/j.amc.2011.12.004, 2879117, ZBL1242.65078.
10.1016/j.amc.2011.12.004
Web of Science® Google Scholar
25 Li F. L., Hu X. Y., and Zhang L., The generalized reflexive solution for a class of matrix equations (AX = C, XB = D), Acta Mathematica Scientia. (2008) 28B, no. 1, 185–193.
Google Scholar
26 Wang Q.-W. and Dong C.-Z., Positive solutions to a system of adjointable operator equations over Hilbert C^*-modules, Linear Algebra and Its Applications. (2010) 433, no. 7, 1481–1489, https://doi.org/10.1016/j.laa.2010.05.023, 2680273.
10.1016/j.laa.2010.05.023
Web of Science® Google Scholar
27 Xu Q., Common Hermitian and positive solutions to the adjointable operator equations AX = C, XB = D, Linear Algebra and Its Applications. (2008) 429, no. 1, 1–11, https://doi.org/10.1016/j.laa.2008.01.030, 2419133, ZBL1153.47012.
10.1016/j.laa.2008.01.030
Web of Science® Google Scholar
28 Zhang F., Li Y., and Zhao J., Common Hermitian least squares solutions of matrix equations and subject to inequality restrictions, Computers & Mathematics with Applications. (2011) 62, no. 6, 2424–2433, https://doi.org/10.1016/j.camwa.2011.07.029, 2831725, ZBL1231.15002.
10.1016/j.camwa.2011.07.029
Web of Science® Google Scholar
29 Chang H.-X., Wang Q.-W., and Song G.-J., (R, S)-conjugate solution to a pair of linear matrix equations, Applied Mathematics and Computation. (2010) 217, no. 1, 73–82, https://doi.org/10.1016/j.amc.2010.04.053, 2672564, ZBL1201.15006.
10.1016/j.amc.2010.04.053
Web of Science® Google Scholar
30 Cheney E. W., Introduction to Approximation Theory, 1966, McGraw-Hill, New York, NY, USA, 0222517.
Google Scholar

Citing Literature

All articles

On the Hermitian R-Conjugate Solution of a System of Matrix Equations

Abstract

1. Introduction

2. R-Conjugate Hermitian Solution to (1.2)

3. The Solution of Optimal Approximation Problem (1.3)

4. The Solution of Problem (1.5)

5. An Algorithm and Numerical Example

Acknowledgments

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

On the Hermitian R-Conjugate Solution of a System of Matrix Equations

Abstract

1. Introduction

2. R-Conjugate Hermitian Solution to (1.2)

3. The Solution of Optimal Approximation Problem (1.3)

4. The Solution of Problem (1.5)

5. An Algorithm and Numerical Example

Acknowledgments

References

Citing Literature

References

Related

Information