An iterative method for solving a kind of constrained linear matrix equations system

An iterative method for solving a kind of constrained linear matrix equations system

Jing Cai^{I, II}; Guoliang Chen^I

^IDepartment of Mathematics, East China Normal University, Shanghai 200062, P.R. China. E-mail: [e-mail]

^IISchool of Science, Huzhou Teachers College, Huzhou Zhejiang 313000, P.R. China. E-mail: [e-mail]

ABSTRACT

In this paper, an iterative method is constructed to solve the following constrained linear matrix equations system: [A₁(X),A₂(X),... ,A_r(X)]=[E1,E2, ... ,E_r ], X ∈ I={X |X= U(X)}, where A_i is a linear operator from C^mxn onto C^pixqi, E_i ∈ C^pixqi, i=1 , 2,..., r , and U is a linear self-conjugate involution operator. When the above constrained matrix equations system is consistent, for any initial matrix X₀ ∈ I, a solution can be obtained by the proposed iterative method in finite iteration steps in the absence of roundoff errors, and the least Frobenius norm solution can be derived when a special kind of initial matrix is chosen. Furthermore, the optimal approximation solution to a given matrix can be derived. Several numerical examples are given to show the efficiency of the presented iterative method.

Mathematical subject classification: 15A24, 65D99,65F30.

Key words: iterative method, linear matrix equations system, linear operator, least Frobenius norm solution, optimal approximation.

1 Introduction

Throughout this paper, we need the following notations. C^mxn denotes the set of m x n complex matrices. For a matrix A ∈C^{m xn}, we denote its transpose, conjugate transpose, trace, Frobenius norm and column space by A^T, A^H, tr(A), ║A║and R(A), respectively. Let I_n and S_n denote the n x n unit matrix and reverse unit matrix respectively. The symbol vec(-) stands for the vec operator, i.e., for A = (a₁, a₂, ...,a_n) ∈C^mxn, where a_i (i = 1, 2 ... ,n) denotes the ith column of A, vec(A) =. In the vector space C^mxn, we define inner product as: 〈X, Y〉 =tr(Y^HX) for all X, Y ∈C^mxn. Two matrices X and Y are said to be orthogonal if 〈X, Y〉= 0.

Let LC^mxn,pxq denote the set of linear operators from C^mxn onto C^pxq. Particularly, when p = m and q = n, LC^mxn,pxq is denoted by LC^mxn. Let τ stand for the identity operator on C^mxn. For A ∈LC^mxn,pxq, its conjugate operator is a linear operator satisfying the following equality:

For example, if is defined as:: X — AXB, then we have : Y → A^HYB^H.

We note that the linear matrix equations, such as the well-known Lyapunov matrix equation XA + A*X = H, Sylvester equation AX ± XB = C, Stein equation X ± AXB = D, AXB + CX^TD = E and [A_xXB_U A₂XB₂] = [Ci, C₂], which have many applications in system theory (see [7-11, 15, 18] for instance), can all be rewritten as the following linear matrix equations system:

where ∈LC^mxn,Pixqi and E_t ∈C^Pixqi, i = 1, 2, ..., r.

Recently, there are many papers considering the matrix equations system (1.1) with various constraints on solutions. For instance, Chu [4], Dai [5] and Chang and Wang [2] considered the symmetric solutions of the matrix equations AX = B, AXB = C and (A^TXA, B^TXB) = (C, D) respectively. Peng et al. [14] and Qiu et al. [16] discussed the reflexive and anti-reflexive solution of the matrix equations A XB = C and AX = B, XC = D, respectively. Li et al. [10], Yuan and Dai [19] and Zhang et al. [20] considered the generalized reflexive and anti-reflexive solutions of the matrix equations (BX = C, XD = E), AXB = D and AX = B. In these literatures, the necessary and sufficient conditions for the consistency of the constrained matrix equations and the explicit expressions of the constrained solutions were derived by using the singular value decomposition (SVD), generalized singular value decomposition (GSVD), canonical correlation decomposition (CCD) or the generalized inverses of matrices. On the other hand, Peng et al. [13] considered an iterative method for symmetric solutions ofthe system of matrix equations A 1XB 1= C 1,A₂XB₂ = C₂. In Huang [9], an iterative method was constructed to obtain the skew-symmetric solution of the matrix equation AXB = C. Dehghan and Hajarian [6] proposed an iterative algorithm to obtain the reflexive solution (namely the generalized centro-symmetric solution) of the matrix equations AYB = E, CYD = F.

It should be remarked that the following common constraints on the solutions of matrix equations (see [3, 12, 14, 19, 20] for more details):

(1) Symmetric (Skew-symmetric) constraint:

X = X^T (X = - X^T);

(2) Centrosymmetric (Centroskew symmetric) constraint:

X = SnXSn( — SnXSn);

(3) Reflexive (Anti-reflexive) constraint:

X = PXP (—PXP);

(4) Generalized reflexive (anti-reflexive) constraint:

X = PXQ (—PXQ);

(5) P orthogonal-symmetric (orthogonal-anti-symmetric) constraint:

(PX)^T= PX(—PX), here P^T = P = P^—1 ≠ I_n and Q^T = Q = Q^—1 ≠ I_n, are all special cases of the following constraint:

where ∈ LC ^xn is a self-conjugate involution operator, i.e., =

Hence it motivates us to consider solving the more general matrix equations system (1.1) with the constraint (1.2) and its associated optimal approximate problem (see the following two problems) by iterative methods, which generalize the main results of [6, 9, 13].

Problem I. Let denote the set of constrained matrices defined by (1.2). For given ∈ LC^mxn,pixqi, E_t ∈ C^pixqi, i = 1, 2, ... , r, find X ∈ such that

Problem II. Let S denote the solution set of Problem I, for given ∈C ^{m xn}, find ∈S, such that

The rest of the paper is organized as follows. In Section 2, e propose an iterative algorithm to obtain a solution or the least Frobenius norm solution of Problem I and present some basic properties of the algorithm. In Section 3, we consider the iterative method for solving Problem II. Some numerical examples are given in Section 4 to show the efficiency of the proposed iterative method. Finally, we put some conclusions in Section 5.

2 Iterative method for solving Problem I

In this section, we first propose an iterative algorithm to solve Problem I, then present some basic properties of the algorithm. We also consider finding the least Frobenius norm solution of Problem I. In the sequel, the least norm solution always means the least Frobenius norm solution.

Algorithm 2.1.

Step 1. Input ∈LC^mxn,Pixqi, E_i∈C^pixqi, i = 1, 2, ..., r, and an arbitrary X₀∈ ;

Step 2. Compute

Step 3. If = 0 or ≠0 but P_k = 0, then stop; else, k := k + 1;

Step 4. Compute

Step 5. Go to Step 3.

Some basic properties of Algorithm 2.1 are listed in the following lemmas.

Lemma 2.1.The sequences {X_i} and {P_i} generated by Algorithm 2.1 are contained in the constraint set .

Proof. We prove the conclusion by induction. For i = 0, by = τ and Algorithm 2.1, we have

i.e., P₀∈ ^.

For i = 1, since x₀∈ , by Algorithm 2.1, we have

Moreover, we have

i.e.,P₁∈ S. Assume that the conclusion holds for i = s (s > 1), i.e., X_s, P_s ∈ S. Then

and

from which we getX_s+₁, P_s+₁∈ .By the principle of induction, we draw the conclusion.

Lemma 2.2. Assume that X* is a solution of Problem I. Then the sequences {X_i}, {} and {P_i} generated by Algorithm 2.1 satisfy the following equality:

Proof. We prove the conclusion by induction. For i = 0, it follows from X*, X₀ ∈ that X* - X₀ ∈ . Then by Algorithm 2.1, we have

For i = 1, by Algorithm 2.1 and Lemma 2.1, we have

Assume that the conclusion holds for i = s(s > 1), i.e., 〈X* — X_s, P_s〉 = , then for i = s + 1, we have

This completes the proof by the principle of induction.

Remark 2.1. Lemma 2.2 implies that if Problem I is consistent, then implies that Pi ≠0. Else if there exists a positive number k such that 0 but Pk =0, then Problem I must be inconsistent. Hence the solvability of Problem I can be determined by Algorithm 2.1 in the absence of roundoff errors.

Lemma 2.3. For the sequences and generated by Algorithm 2.1, it follows that

Proof. By Algorithm 2.1, Lemma 2.1 and , we have

This completes the proof.

Lemma 2.4. Assume that Problem I is consistent. If there exists a positive integer k such that for all i = 0, 1, 2, ... , k, then the sequences and generated byAlgorith 2.1 satisfy

and, (i, j = 0, 1, 2, ..., k, i = j). (2.3)

Proof. Note that holds for arbitrary matrices A and B. We only need to prove the conclusion holds for all 0 < j < i < k. For k = 1, by Lemma 2.3, we have

and

Assume that and = 0 for all 0 <j < s, 0 < s <k, we shall show that = 0 and hold for all 0 < j < s + 1, 0 < s + 1 < k.

By the hypothesis and Lemma 2.3, for the case where 0 < j < s, we have

and

For the case j = s, we have

and

Then by the principle of induction, we complete the proof.

Remark 2.2. Lemma 2.4 implies that when Problem I is consistent, the sequences P₀, P₁, P₂ ... are orthogonal to each other in the finite dimension matrix space . Hence there exists a positive integer t < mn such that P_t = 0. According to Lemma 2.2, we have . Hence the iteration will be terminated in finite steps in the absence of roundoff errors.

Next we consider finding the least norm solution of Problem I. The following lemmas are needed for our derivation.

Lemma 2.5 [1]. Suppose that the consistent system of linear equations Ax =b has a solution x* ∈R (A^H), then x*is the unique least norm solution of the syste oflinear equations.

Lemma 2.6 [8]. For ∈ LC^mxn,pxq, there exists a unique matrix M ∈ C^pqxmn, such that vec ((X)) = M vec(X) for all X ∈ C^mxn.

According to Lemma 2.6 and the definition of conjugate operator, one can easily obtain the following corollary.

Corollary 2.1. Let and M be the same as those in Lemma 2.4, and be the conjugate operator of ,then vec ((Y)) = M^H vec(Y) for all Y ∈ C^pxq.

Theorem 2.1. Assume that Problem I is consistent. If we choose the initial matrix , where H is an arbitrary matrix in C ^plxql, or ore especially, let X₀ = 0, then the solution X^* obtained by Algorith 2. 1 is the least nor solution.

Proof. Consider the following matrix equations system:

If Problem I has a solution X, then X must be a solution of the system (2.4). Conversely, if the system (2.4) has a solution X, let , then it is easy to verify that is a solution of Problem I. Therefore, the solvability of Problem I is equivalent to the system (2.4).

If we choose the initial matrix , where H is an arbitrary matrix in C^plxql, by Algorithm 2.1 and Remark 2.2, we can obtain the solution X* of Problem I within finite iteration steps, which can be represented as . Since the solution set of Problem I is a subset of that of the system (2.4), next we shall prove that X* is the least norm solution of the system (2.4), which implies that X^* is the least norm solution of Problem I.

Let M_l and N be the matrices such that vec((X)) = M_l vec(X) and vec((X)) = N vec(x)for all X ∈ C^mxn, l = 1, 2, ... , r. Then the system (2.4) is equivalent to the following system of linear equations:

Note that

Then it follows from Lemma 2.5 thatX*is the least norm solution ofthe system (2.4), which is also the least norm solution of Problem I.

3 Iterative method for solving Problem II

In this section, we consider iterative method for solving the matrix nearness Problem II. Let S denote the solution set of Problem I, for given ∈ C ^mxn and arbitrary X ∈ S, we have

Note that

which implies . Then (3.1) reduces to

Hence is equivalent to , where

Let and . Then is a solution of the following constrained matrix equations system:

Hence Problem II is equivalent to finding the least norm solution of above constrained matrix equations system.

Based on the analysis above, we summarize that once the unique least norm solution of above constrained consistent matrix equations system is obtained by applying Algorithm 2.1, the unique solution of Problem II can be obtained, where

4 Numerical examples

In this section, we shall give some numerical examples to illustrate the efficiency of Algorithm 2.1. All the tests are performed by MATLAB 7.6 with machine precision around 10^—16. Because of the influence of roundoff errors, we regard a matrix X as zero matrix if ║X║ < 1.0e — 010.

Example 4.1. Find the least norm symmetric solution of the following matrix equations system:

where

If we define : X → A^T X + X^T A and : X → BXB^T, then we have : Y → AY + AY^T and : Y → B^TYB. Here we define : X → X^T. Let X 0= 0, then by using Algorithm 2.1 and iterating 16 steps, we obtain the least norm symmetric solution of the system (4.1) as follows:

with = 1.4901e - 018.

Example 4.2. Let S_Edenote the solution set of the matrix equations system (4.1) with symmetric constraint, for given

find , such that .

In order to find the optimal approximation solution to the matrix , let = , , and .

By applying Algorithm 2.1 to the new matrix equations system:

with symmetric constraint, and choosing the initial matrix , we obtain its unique least norm symmetric solution as follows:

with

Then we have

and

5 Conclusions

In this paper, an iterative algorithm is constructed to solve a kind of constrained matrix equations system, i.e., the matrix equations system (1.1) with the constraint (1.2), and its associated optimal approximation problem. By this algorithm, for any initial matrix X0satisfying the constraint (1.2), a solution X* can be obtained in finite iteration steps in the absence of roundoff errors, and the least norm solution can be obtained by choosing a special kind of initial matrix.

In addition, using this iterative method, the optimal approximation solution to a given matrix can be derived by first finding the least norm solution of a new corresponding constrained matrix equations system. The given numerical examples show that the proposed iterative algorithm is quite efficient.

Acknowledgement. Research supported by Shanghai Science and Technology Committee (No. 062112065) and Natural Science Foundation of Zhejiang Province (No. Y607136).

Received: 13/III/09.

Accepted: 04/VI/09.

#CAM-77/09.

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Publication Dates

• Publication in this collection
  05 Nov 2009
• Date of issue
  2009

History

• Accepted
  04 June 2009
• Received
  13 Mar 2009