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

The iterative method is presented for obtaining the centrally symmetric (centrally antisymmetric) matrix pair (X, Y) solutions of the generalized coupled Sylvester-conjugate matrix equations A 1 X + B 1 Y = D 1 XE 1 + F 1 , A 2 Y + B 2 X = D 2 YE 2 + F 2 . On the condition that the coupled matrix equations are consistent, we show that the solution pair (X∗, Y∗) 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.

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.
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.

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].
For all real constants  1 ,  2 , 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 ‖‖ 2  = tr(  ) and    is a Hermite matrix.Then by the knowledge of algebra, we know that tr(  ) is real; hence, tr(  ) = Re[tr(  )].
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].
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). ) , set  := 1.
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 7. Suppose that the system of matrix equations (4) is consistent; let ( * ,  * ) be an arbitrary solution pair of (4).
Then for any initial matrices  1 ∈ Proof.The conclusion is accomplished by mathematical induction.
Firstly, we notice that the sequences pair (  ,   ), ( = 1, 2, . ..) generated by Algorithm 3 are all central symmetric matrices since initial matrix pair ( 1 ,  1 ) is centrally symmetric matrix.Then for  = 1, it follows from Algorithm 3 that In the same way, we can get This shows that That is, (38) holds, for  = 1.Assume (38) holds, for  = .For  =  + 1, it follows from the updated formulas of  +1 ,  +1 that Then On the other hand, we have Therefore, by (20) we get Hence, the proof is completed.
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  1 ∈  × ,  1 ∈  × , an exact solution of (4) can be derived at most 2 + 1 iteration steps by Algorithm 3.
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.For a rigorous proof of this lemma above the reader is referred to [46,48]. 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).
Obviously, if  ∈ R(), then  ∈ R((1/2)), where  is a matrix.So, from the above analysis, we can get the result.
, and the linear matrix equation  =  has a solution  * ∈  4 2  , where , then  * is the unique least Frobenius norm solution of  =  in (59).
Proof.If  1 ,  1 has the form of (60), (61), respectively, by step 2 of Algorithm 3, we have we have Re (69) In the same way, we can prove that for all solutions (, ) of (4).Hence ( * ,  * ) is the unique least Frobenius norm solution of (4).

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.

Conclusion
Iterative method is proposed to solve the generalized coupled Sylvester-conjugate linear matrix equations  1  +  1  =  1  1 +  1 ,  2  +  2  =  2  2 +  2 for centersymmetry (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 centersymmetry (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.

Lemma 11 .
Suppose  ∈  × ,  ∈   , and the linear matrix equation  =  has a solution  * ∈ R(  ); then  * is the unique least Frobenius norm solution of  = .

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

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