ABSTRACT

In this paper, we discuss least squares symmetrizable solutions of matrix equations (AX = B, XC = D) and its optimal approximation solution. With the matrix row stacking, Kronecker product and special relations between two linear subspaces are topological isomorphism, and we derive the general solutions of least squares problem. With the invariance of the Frobenius norm under orthogonal transformations, we obtain the unique solution of optimal approximation problem. In addition, we present an algorithm and numerical experiment to obtain the optimal approximation solution.

1. Introduction

The matrix equations (AX = B, XC = D), where A, B, C, D are usually given by experiments, have a long history [1]. Many authors considered these matrix equations. For example, Mitra [2,3], Chu [4] discussed its unconstraint solutions with the generalized inverse of matrix and the singular value decomposition (SVD), respectively. In recent years, many authors considered its constraint solutions. A series of meaningful results were achieved [1,5- 10]. The methods in these papers are mainly the generalized inverse of matrix and special properties of finite dimensional vector spaces [1,5,6], the decomposition of matrix or matrix pairs [7,8] and the special properties of constraint matrices [9,10]. However, the least squares symmetrizable solutions for these matrix equations have not been considered. The purpose of this paper is to discuss its least squares symmetrizable solutions with the matrix row stacking, Kronecker product and special relations between two linear subspaces which are topological isomorphism because the structure of symmetrizable matrices can not be found and the methods applied in [1-10] can not solve the problem in this paper. The background for introducing the definition of symmetrizable matrices is to get “symmetric” matrices from nonsymmetric matrices [11] because of nice properties and multi-areas of applications of symmetric matrices. For example, Sun [12] introduced the definition of positive definite symmetrizable matrices to study an efficient algorithm for solving the nonsymmetry second-order elliptic discrete systems.

Throughout this paper we use some notations as follows. Let be the set of all real matrices and denote;, and are the set of all orthogonal, symmetric and skewsymmetric matrices, respectively., and represent the range, the transpose and the Moore-Penrose generalized inverse of A, respectively.

I_n denotes the identity matrix of order n. For, , denotes Kronecker product of matrix A and B; denotes the inner product of matrix A and B. The induced matrix norm is called Frobenius norm, i.e., then is a Hilbert inner product space.

Definition 1. A real matrix A is called a symmetrizable (skew-symmetrizable) matrix if A is similar to a symmetric (skew-symmetric) matrix A. The set of symmetrizable (skew-symmetrizable) matrices is denoted by .

From Definition 1, it is easy to prove that if and only if there exists a nonsingular matrix W and a symmetric (skew-symmetric) matrix such that

We now introduce the following two special classes of subspaces in.

,

.

It is easy to see that if W is a given nonsingular matrix, then and are two closed linear subspace of. In this paper, we suppose that W is a given nonsingular matrix and. We will consider the following problems.

Problem I. Giving, , find such that

.

Problem II. Given, find such that

where is the solution set of Problem I.

In this paper, if C = 0, D = 0 in Problem I, then Problem I becomes Problem I of [13]. Peng [13] studied the least squares symmetrizable solutions of the matrix equation AX = B with the singular value decomposition of matrix. The method applied in [13] can not solve Problem I in this paper. In this paper, we first take matrix equations (AX = B, XC = D) into linear equations with matrix row stacking and Kronecker product. Then we obtain an orthogonal basis-set for with special relations between two linear subspaces which are topological isomorphism. Based on these results, we obtain the general expression of Problem I.

This paper is organized as follows. In Section 2, we first discuss the matrix row stacking methods, Kronecker product of matrix and relations between and. Then we obtain the general solutions of Problem I. In Section 3, we derive the solution of Problem II with the invariance of Frobenius norm under orthogonal transformations. In the end, we give an algorithm and numerical experiment to obtain the optimal approximation solution.

2. The Solution Set of Problem I

At first, we discuss the matrix row stacking methods, Kronecker product of matrix and relations between two linear subspaces which are topological isomorphism.

For any, let denote an ordered stack of the row of A from upper to low stacking with the first row, i.e.

, (2.1)

where denotes the ith row of A. For any vector, let denote the following matrix containing all the entries of vector x.

, (2.2)

where denotes the elements from i to j of vector. From (2.1), we can derive the following two linear subspaces of.

, (2.3)

.

Lemma 1. [14] If, , , then

. (2.4)

For any, let

, i.e..   (2.5)

It is no difficult to prove that mapping is a topological isomorphism mapping from to. According to (2.1) and (2.5), it is easy to derive the following mapping from linear subspaces to.

i.e.

(2.6)

It is also easy to prove that mapping is a topological isomorphism mapping from to

. It is clear that the dimension of

is. This implies that the dimension of and are also. In this paper, let.

Lemma 2. If is an orthonormal basisset for, and let, then the following relations hold.

(2.7)

From the definition of the orthonormal basis-set, it is easy to prove Lemma 2, so the proof is omitted.

For any matrix, if let denote the vector of coordinates of with respect to the basis-set, then combining (2.4) and (2.7), we have

(2.8)

Moreover, for any, the following conclusion holds.

. (2.9)

The orthonormal basis-set for can be obtained by the following calculation procedure.

Calculation procedure

Step 1. Input a basis-set for.

Step 2. According to (2.1), compute , and obtain a basis-set for.

Step 3. Input a nonsingular matrix W, compute

and obtain a basis-set for.

Step 4. Compute the QR decomposition of matrix

and obtain an orthonormal basis-set for.

Lemma 3. If, , then the general solutions of least squares problem

is

.

With the singular value decomposition of matrix, it is easy to prove this lemma. So the proof is omitted.

Theorem 1. Giving, , and let

, (2.11)

then the general solutions of Problem I is

(2.12)

Proof. From Lemma 1, we have

This implies that finding such that if and only if finding such that

, (2.13)

where. From Lemma 3, the general solutions of (2.13) is

(2.14)

Combining (2.13) and (2.14) gives (2.12).         □

3. The Solution of Problem II

Let be the solution set of Problem I. From (2.12), it is easy to see that is a nonempty closed convex set. So we claim that for any given, there exists the unique optimal approximation for Problem II.

Theorem 2. If given, , , then Problem II has a unique solution. Moreover, can be expressed as

(3.1)

where are denoted by (2.11).

Proof. Choose such that. Combining the invariance of the Frobenius norm under orthogonal transformations, (2.12) and (2.14), we have

Let, , it is clear that are orthogonal projection matrices satisfying. Hence, we have

It is easy to prove that. This implies that

(3.2)

The solution of (3.2) is

. (3.3)

Substituting (3.3) to (2.12) gives (3.1).            □

From Theorem 2, we can design the following algorithm to obtain the optimal approximate solution.

Algorithm

1) Input.

2) Input a basis-set for.

3) According to the calculation procedure before, compute and obtain an orthonormal basis-set for.

4) Let, compute A₀, B₀ from (2.11).

5) Compute from the second equation of (3.1).

6) According to the first equation of (3.1), calculate.

Example

1) Input as follows.

,

,

,

,

.

2) Input a basis-set for as follows.

3) According to the calculation procedure, we obtain an orthonormal basis-set for

as follows.

4) Using the software “MATLAB”, we obtain the unique solution of Problem II.

.

4. Conclusion

In this paper, we first derive the least squares symmetrizable solutions of matrix equations (AX = B, XC = D) with the matrix row stacking and the theory of topological isomorphism, i.e. Theorem 1. Then we give the unique optimal approximation solution, i.e. Theorem 2. Based on Theorem 1 and 2, we design an algorithm to find the optimal approximation solution. Compare to [1-10], this paper has two important achievements. One is we apply the topological isomorphism theory to obtain the least squares symmetrizable solutions of matrix equations (AX = B, XC = D), and provide a method to solve the matrix equation, where the construct of constraint matrix can not be found. The other is we present a stable calculation procedure to obtain an orthonormal basis-set for , and solve the key problem of the algorithm.

5. Acknowledgements

The authors are very grateful to the referee for their valuable comments, and also thank for his helpful suggestions.

This research was supported by National natural Science Foundation of China (31170532).

REFERENCES