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The reflexive and Hermitian reflexive solutions of the generalized Sylvester-conjugate matrix equation.

1 Introduction

In this paper, we denote the set of all m x n complex matrices by [C.sup.m x n] and the identity matrix with the appropriate size by I. The symbols [A.sup.T], [[bar].A], [A.sup.H] and tr(A) mean the transpose, conjugate, conjugate transpose and trace of a matrix A, respectively. Re(a) denotes the real part of number a. The inner product [<.,.>.sub.r] in [C.sup.m x n] over the field R is defined as follows:

[<A, B>.sub.r] = Re(tr([B.sup.H]A)) for A, B [member of] [C.sup.m x n],

that is [<A, B>.sub.r] is the real part of the trace of [B.sup.H]A. It is can be shown that ([C.sup.m x n], r, [<.,.>.sub.r]) is a Hilbert inner product space. The induced matrix norm is [parallel]A[parallel] = [square root of ([<A,A>.sub.r])] = [square root of (Re(tr([A.sup.H]A))]. which is the Frobenius norm [1,47,48]. A matrix P [member of] [C.sup.n x n] is called generalized reflection matrix if P = [P.sup.H] and [PP.sup.H] = I. Throughout, we always suppose that P and Q are a given generalized reflection matrices. If A = PAP (A = [A.sup.H] = PAP) then A is called a reflexive (Hermitian reflexive) matrix with respect to P. [C.sup.n x n.sub.r](P) ([HC.sup.n x n.sub.r](P)) denotes the set of order n reflexive (Hermitian reflexive) matrices with respect to P. Due to that I is a generalized reflection matrix, any n x n complex (Hermitian) matrix is also a reflexive (Hermitian reflexive) matrix with respect to I. The reflexive matrices (namely the generalized centro-symmetric matrices) have practical applications in many areas such as the numerical solution of certain differential equations [2, 5], pattern recognition [7], Markov processes [43], various physical and engineering problems [8, 6] and so on (e.g. [3, 27, 45, 46]). Chen [4] proposed three applications of reflexive matrices obtained from the altitude estimation of a level network, an electric network and structural analysis of trusses. The symmetric Toeplitz matrices, an important subclass of the class of Hermitian reflexive matrices, appear naturally in digital signal processing applications and other areas [18]. The problem of finding solutions of linear matrix equations arises in a variety of engineering, mathematics and physics problems [9, 15, 30, 31, 32, 33, 35]. Therefore, in recent years much attention has focused on studying the solutions of linear matrix equations. In [36, 37, 38, 39, 40, 42] several quaternion matrix equations were studied. In [19, 21, 23], by extending the well-known Jacobi and Gauss-Seidel iterations, Ding and Chen presented some efficient iterative algorithms based on the hierarchical identification principle [20, 22] for solving the generalized Sylvester matrix equations and general coupled matrix equations. In [44,24], this efficient approach was also applied for other general matrix equations. In [49,50], Zhou and Duan studied the solution of the generalized Sylvester matrix equations. In [51], Zhou et al. analyzed the computational complexity of the Smith iteration and its variations for solving the Stein matrix equation X - AXB = C. By extending the idea of conjugate gradient (CG) method, Dehghan and Hajarian proposed some efficient iterative methods to solve Sylvester and Lyapunov matrix equations [17, 10, 11, 12, 13, 14]. Thiran et al. [34] investigated a rational approximation problem in connection with the convergence analysis of the ADI iterative method applied to the Stein matrix equation. Jiang and Wei [28] obtained explicit solutions of the Stein matrix equation and Stein-conjugate matrix equation X - A[bar.X]B = C by the method of characteristic polynomial and a method of real representation of a complex matrix respectively. It is known that solving complex matrix equations can be very difficult and it is sufficiently complicated. This paper is concerned with the reflexive (Hermitian reflexive) solution pair [X, Y] of the generalized Sylvester-conjugate matrix equation

[A.sub.1]X[B.sub.1] + [A.sub.2][[bar].X][B.sub.2] + [C.sub.1]Y[D.sub.1] + [C.sub.2][[bar].Y][D.sub.2] = E, (1.1)

where [A.sub.1],[A.sub.2] [member of] [C.sup.p x n], [B.sub.1],[B.sub.2] [member of] [C.sup.n x q], [C.sub.1], [C.sub.2] [member of] [C.sup.p x m], [D.sub.1], [D.sub.2] [member of] [C.sup.m x q], E [member of] [C.sup.p x q] are known matrices and X [member of] [C.sup.n x n.sub.r](P), Y [member of] [C.sup.m x m.sub.r](Q) (X [member of] [HC.sup.n x n.sub.r](P), Y [member of] [HC.sup.m x m.sub.r](Q)) are unknown matrices to be determined. The reflexive and Hermitian reflexive solutions of the matrix equation (1.1) have not been dealt with yet. This matrix equation includes various linear matrix equations such as Lyapunov, Sylvester, Stein, Yakubovich, Kalman-Yakubovich, homogeneous (nonhomogeneous) Yakubovich-conjugate matrix equations as special cases. Hence the generalized Sylvester-conjugate matrix equation (1.1) can play an important role in control theory and can be used to achieve pole assignment, robust pole assignment and observer design for descriptor linear systems [26]. The rest of the paper is organized as follows. In Section 2, by extending the Jacobi and Gauss-Seidel iterations we propose two GI methods to solve (1.1) over reflexive and Hermitian reflexive matrices. Theoretical analysis shows that the proposed methods converge to the reflexive and Hermitian reflexive solutions of (1.1) for any initial reflexive and Hermitian reflexive matrices, respectively. Finally, two numerical examples are given to illustrate the effectiveness of the proposed methods in Section 3.

2 Main results

In this section we propose two iterative methods for finding the reflexive and Hermitian reflexive solutions of (1.1) respectively and their convergence analysis is also given.

In [21, 29], some iterative methods were presented to solve Sylvester matrix equations over real matrices. In the methods [21, 29], matrix inversion is required in the first iteration. These methods may turn out to be numerically expensive and are not practical for equations of large systems. The purpose in this paper is to obtain two iterative methods without any inverse for solving the linear matrix equation (1.1) over the reflexive and Hermitian reflexive matrices. First, it is known that the solvability of linear matrix equation (1.1) over the reflexive (Hermitian reflexive) matrix pair [X, Y] is equivalent to the following system of matrix equations:

{[A.sub.1]X[B.sub.1] + [A.sub.2][[bar].X][B.sub.2] + [C.sub.1]Y[D.sub.1] + [C.sub.2][[bar].Y][D.sub.2] = E, [A.sub.1]PXP[B.sub.1] + [A.sub.2]P[bar.X]P[B.sub.2] + [C.sub.1]QYQ[D.sub.1] + [C.sub.2]Q[bar.Y]Q[D.sub.2] = E, (2.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

Now consider a linear algebraic system of equations

Ax = b, (2.3)

which A is the coefficient matrix; and b and x are, respectively, the known right hand side and the solution to be sought. Also suppose that

A = D - E - F, (2.4)

in which D is the diagonal of A, -E its strict lower part, and -F its strict upper part. It is always assumed that the diagonal entries of A are all nonzero. To solve the linear system (2.3), the Jacobi and the Gauss-Seidel iterations are both of the form

Mx(k + 1) = Nx(k) + b = (M - A)x(k) + b, (2.5)

in which

A = M - N, (2.6)

is a splitting of A, with M = D for Jacobi, M = D - E for forward Gauss-Seidel, and M = D - F for backward Gauss-Seidel. Here by extending the Jacobi and the Gauss-Seidel iterations (2.5) and by applying the hierarchical identification principle for (2.1) and (2.2), respectively, we can obtain the GI methods described in the following:

Algorithm 1. To solve (1.1) over reflexive matrix pair [X, Y]:

Given an initial reflexive matrix pair [X(1),Y(1)] with X(1) [member of] [C.sup.n x n.sub.r](P) and Y(1) [member of] [C.sup.m x m.sub.r](Q);

For k:=1,2,.... until convergence do;

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Algorithm 2. To solve (1.1) over Hermitian reflexive matrix pair [X, Y]:

Given an initial Hermitian reflexive matrix pair [X(1), Y(1)] with X(1) [member of] [HC.sup.n x n.sub.r](P) And Y(1) [member of] [HC.sup.m x m.sub.r](Q);

For k:=1,2,.... until convergence do;

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From Algorithm 1 (2), we can see that X(k) [member of] [C.sup.n x n.sub.r](P) and Y(k) [member of] [C.sup.m x m.sub.r](Q) (X(k) [member of] [HC.sup.n x n.sub.r](P) and Y(k) [member of] [HC.sup.m x m.sub.r](Q)) for k = 1,2,.... Now we present the main results of this paper.

Theorem 2.1. If the generalized Sylvester-conjugate matrix equation (1.1) has a unique reflexive solution pair [[X.sup.*], [Y.sup.*]], then the iterative solution pair [X(k), Y(k)] given by the Algorithm 1 converges to [[X.sup.*], [Y.sup.*]], i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any initial reflexive matrix pair [X(1), Y(1)].

Proof. First we define the estimation error matrices as:

[[xi].sub.1] (k) = X(k) - [X.sup.*] and [[xi].sub.2] (k) = Y(k) - [Y.sup.*].

It is obvious that [[xi].sub.1] (k) [member of] [C.sup.n x n.sub.r] and [[xi].sub.2](k) [member of] [C.sup.m x m.sub.r](Q) for k = 1,2,.... By using the above error matrices and Algorithm 1, we can obtain

R(k) = -[A.sub.1] [[xi].sub.1](k)[B.sub.1] - [A.sub.2][bar.[[xi].sub.1](k)][B.sub.2] - [C.sub.1] [[xi].sub.2](k)[D.sub.1] - [C.sub.2][bar.[[xi].sub.2](k)][D.sub.2],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

By taking the norm of both sides of (2.7) and using following facts for two square complex matrices A, B and the generalized reflection matrix P

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly to the above, we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Define the nonnegative definite function [] (k) by:

[xi](k) = [[parallel][[xi].sub.1](k)[parallel].sup.2] + [[parallel][[xi].sub.2](k)[parallel].sup.2].

By the previous results, this function can be computed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If the convergence factor [mu] is chosen to satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.10)

then we can conclude that

[[infinity].summation over (i=1)][parallel][A.sub.1][[xi].sub.1](i)[B.sub.1] + [A.sub.2][bar.[[xi].sub.1](i)][B.sub.2] + [C.sub.1][bar.[[xi].sub.2](i)][D.sub.1] + [C.sub.2][bar.[[xi].sub.2](i)][D.sub.2][[parallel]sup.2] < [infinity]. (2.11)

Because if

[[infinity].summation over (i=1)][parallel][A.sub.1][[xi].sub.1](i)[B.sub.1] + [A.sub.2][bar.[[xi].sub.1](i)][B.sub.2] + [C.sub.1][bar.[[xi].sub.2](i)][D.sub.1] + [C.sub.2][bar.[[xi].sub.2](i)][D.sub.2][[parallel]sup.2] = [infinity],

then by considering (2.10) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.12)

Now it is obvious that (2.12) contradicts [xi](k) [greater than or equal to] 0.

The necessary condition of the series convergence (2.11) implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If we define matrices [M.sub.1] := [lim.sub.i[right arrow][infinity]] [[xi].sub.1] (i) and [M.sub.2] := [lim.sub.i[right arrow][infinity]] [[xi].sub.2] (i) then the above relation can be written by

[A.sub.1][M.sub.1][B.sub.1] + [A.sub.2][bar.[M.sub.1]][B.sub.2] + [C.sub.1][M.sub.2][D.sub.1] + [C.sub.2][bar.[M.sub.2]][D.sub.2] = 0. (2.13)

It follows from (1.1) (or (2.13)) has a unique reflexive solution pair that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The proof is finished.

Similarly to the proof of Theorem 2.1, we can prove the following theorem.

Theorem 2.2. If the generalized Sylvester-conjugate matrix equation (1.1) has a unique Hermitian reflexive solution pair [[X.sup.*], [Y.sup.*]], then the iterative solution pair [X(k), Y(k)] given by the Algorithm 2 converges to [[X.sup.*], [Y.sup.*]], i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any initial Hermitian reflexive matrix pair [X(1), Y(1)].

3 Numerical experiments

This section gives two numerical experiments to illustrate the convergence behaviors of both Algorithms 1 and 2. All codes were written in Matlab. All the experiments were performed on a PC of Intel Pentium 2.0 GHz.

Example 3.1. In this example we consider the matrix equation

X + A[[bar].X]B = C, (3.1)

with the following parameters

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It can be verified that this matrix equation is consistent over reflexive matrices and has the reflexive solution [X.sup.*] [member of] [C.sup.5x5.sub.r](P), that is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We apply Algorithm 1 with the initial reflexive matrix X (1) = 0 to calculate {X(k)}. The derived results are displayed in Figure 1, where

r(k) = [log.sub.10] [parallel]R(k)[parallel] (residual) and [delta](k) = [log.sub.10] [[[parallel]X(k) -[X.sup.*][parallel]]/[parallel][X.sup.*][parallel]] (relative error).

We can easily see that r(k), [delta](k) decrease and converge to zero as k increases. In [28], the solution of complex matrix equation (3.1) was obtained by the method of characteristic polynomial and a method of real representation of a complex matrix respectively. Because of characteristic polynomial, the method [28] may turn out to be numerically expensive and is not practical for equations of large systems.

Example 3.2. Consider the matrix equation

AX + YB = C, (3.2)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We can verify that the matrix equation (3.2) is consistent over Hermitian reflexive matrices and has Hermitian reflexive solution pair [[X.sup.*], [Y.sup.*]] with [X.sup.*], [Y.sup.*] [member of] [HC.sup.5x5.sub.r](P) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By Algorithm 2 for (3.2) with the the initial matrix pair [X(1),Y(1)] = [0, 0] we obtain the sequences {X(k)} and {Y(k)}. The obtained results are presented in Figure 2. From Figure 2, we can see that Algorithm 2 is effective.

4 Conclusions

In this work, we have proposed Algorithms 1 and 2, respectively, for computing the reflexive and Hermitian reflexive solutions of (1.1). We have proven that Algorithms 1 and 2 always converge to the reflexive and Hermitian reflexive solutions for any initial reflexive and Hermitian reflexive matrices, respectively. Moreover, we have presented two numerical examples to test the performance of the proposed algorithms.

Acknowledgments

The authors would like to express their heartfelt thanks to the anonymous referee for his\her useful comments and constructive suggestions which substantially improved the quality and presentation of this paper. The authors are also very much indebted to Professor Karel in't Hout (Editor) for his valuable suggestions, generous concern and encouragement during the review process of this paper.

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Masoud Hajarian * Mehdi Dehghan

Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, General Campus, Evin, Tehran 19839, Iran Email: m_hajarian@sbu.ac.ir; mhajarian@aut.ac.ir; masoudhajarian@gmail.com

Department of Applied Mathematics Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No.424, Hafez Avenue, Tehran 15914, Iran Email: mdehghan@aut.ac.ir, mdehghan.aut@gmail.com

* Corresponding author. Received by the editors in July 2011--In revised form in June 2012. Communicated by K. In't Hout. 2010 Mathematics Subject Classification : 15A24; 65F10; 65F30.
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