A New Method for the Bisymmetric Minimum Norm Solution of the Consistent Matrix Equations ,
We propose a new iterative method to find the bisymmetric minimum norm solution of a pair of consistent matrix equations , . The algorithm can obtain the bisymmetric solution with minimum Frobenius norm in finite iteration steps in the absence of round-off errors. Our algorithm is faster and more stable than Algorithm 2.1 by Cai et al. (2010).
Let denote the set of real matrices. A matrix is said to be bisymmetric if for all , . Let denote real bisymmetric matrices. For any , , , , and represent the transpose, trace, Frobenius norm, and Euclidean norm of , respectively. The symbol stands for the vec operator; that is, for , where denotes the th column of , . Let represent the inverse operation of vec operator. In the vector space , we define the inner product as for all . Two matrices and are said to be orthogonal if . Let denote the reverse unit matrix where is the th column of unit matrix ; then .
In this paper, we discuss the following consistent matrix equations: where , , , , , and are given matrices, and is unknown bisymmetric matrix to be found.
Research on solving a pair of matrix equations , has been actively ongoing for the past 30 or more years (see details in [1–6]). Besides the works on finding the common solutions to the matrix equations , , there are some valuable efforts on solving a pair of the matrix equations with certain linear constraints on solution. For instance, Khatri and Mitra  derived the Hermitian solution of the consistent matrix equations , . Deng et al.  studied the consistent conditions and the general expressions about the Hermitian solutions of the matrix equations and designed an iterative method for its Hermitian minimum norm solutions. Peng et al.  presented an iterative method to obtain the least squares reflexive solutions of the matrix equations , . Cai et al. [10, 11] proposed iterative methods to solve the bisymmetric solutions of the matrix equations , .
In this paper, we propose a new iterative algorithm to solve the bisymmetric solution with the minimum Frobenius norm of the consistent matrix equations , , which is faster and more stable than Cai’s algorithm (Algorithm 2.1) in .
The rest of the paper is organized as follows. In Section 2, we propose an iterative algorithm to obtain the bisymmetric minimum Frobenius norm solution of (1) and present some basic properties of the algorithm. Some numerical examples are given in Section 3 to show the efficiency of the proposed iterative method.
2. A New Iterative Algorithm
Firstly, we give the following lemmas.
Lemma 1 (see ). There is a unique matrix such that for all . This matrix depends only on the dimensions and . Moreover, is a permutation matrix and .
Lemma 2. If are orthogonal to each other, then there exists a positive integer such that .
Proof. If there exists a positive integer such that , then Lemma 2 is proved.
Otherwise, we have , , and are orthogonal to each other in the -dimension vector space of . So form a set of orthogonal basis of .
Hence can be expressed by the linear combination of . Denote in which , . Then From and , , we have ; that is, This completes the proof.
Lemma 3. A matrix if and only if .
Lemma 4. If , then .
Next, we review the algorithm proposed by Paige  for solving the following consistent problem: with given .
Algorithm 5 (Paige algorithm). (i) Initialization
(ii) Iteration. For , until convergence, do(a); ;(b); ;(c); (d); (e); (f); (g).
It is well known that if the consistent system of linear equations has a solution , then is the unique minimum Euclidean norm solution of . It is obvious that generated by Algorithm 5 belongs to and this leads to the following result.
Theorem 6. The solution generated by Algorithm 5 is the minimum Euclidean norm solution of (5).
If and are generated by Algorithm 5, then , (see details in ), in which If we denote where is the approximation solution obtained by Algorithm 5 after the th iteration, it follows that (see details in ). So we have in which .
Now we derive our new algorithm, which is based on Paige algorithm.
Noting that is the bisymmetric solution of (1) if and only if is the bisymmetric solution of the following linear equations: Furthermore, suppose (10) is consistent; let be a solution of (10). If is a bisymmetric matrix, then is a bisymmetric solution of (1); otherwise we can obtain a bisymmetric solution of (10) by .
And so, the vector form of , , , and in Algorithm 5 can be rewritten as matrix form. Then we now propose the following matrix-form algorithm.
Algorithm 7. (i) Initialization ; ; ; ; ; ; ; ; ; .
(ii) Iteration. For , until convergence, do(a); (b); (c); ; ;(d);(e); ; ; ;(f);(g).
Remark 8. The stopping criteria on Algorithm 7 can be used as where is a small tolerance.
Some basic properties of Algorithm 7 are listed in the following theorems.
Theorem 11. The iteration of Algorithm 7 will be terminated in at most steps in the absence of round-off errors.
Proof. By (8) and (11), we have by simple calculation that
in which , and , where is the approximation solution obtained by Algorithm 7 after the th iteration.
By Lemma 1, we have that where and are permutation matrices. For simplicity, we denote , . Then .
If we let , then we have by (9) that are orthogonal to each other in . By Lemma 2, there exists a positive integer such that . Hence that is, the iteration of Algorithm 7 will be terminated in at most steps in the absence of round-off errors.
3. Numerical Examples
In this section, we use some numerical examples to illustrate the efficiency of our algorithm. The computations are carried out at PC computer, with software MATLAB 7.0. The machine precision is around .
We stop the iteration when .
Example 12. Given matrices , , , , , and as follows: then (1) is consistent, for one can easily verify that it has a bisymmetric solution: We choose the initial matrix , then using Algorithm 7 and iterating 13 steps, we have the unique bisymmetric minimum Frobenius norm solution of (1) as follows:with .
This research was supported by the Natural Science Foundation of China (nos. 10901056, 11071079, and 11001167), the Shanghai Science and Technology Commission “Venus” Project (no. 11QA1402200), and the Natural Science Foundation of Zhejiang Province (no. Y6110043).
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