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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 423605, 9 pages
The Hermitian -Conjugate Generalized Procrustes Problem
1Department of Applied Mathematics, Shanghai Finance University, Shanghai 201209, China
2School of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin 541004, China
3Department of Mathematics, Shanghai University, Shanghai 200444, China
Received 10 May 2013; Accepted 14 August 2013
Academic Editor: Masoud Hajarian
Copyright © 2013 Hai-Xia Chang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the Hermitian -conjugate generalized Procrustes problem to find Hermitian -conjugate matrix such that is minimum, where , , , and (, ) are given complex matrices, and and are positive integers. The expression of the solution to Hermitian -conjugate generalized Procrustes problem is derived. And the optimal approximation solution in the solution set for Hermitian -conjugate generalized Procrustes problem to a given matrix is also obtained. Furthermore, we establish necessary and sufficient conditions for the existence and the formula for Hermitian -conjugate solution to the linear system of complex matrix equations , , ( and are positive integers). The representation of the corresponding optimal approximation problem is presented. Finally, an algorithm for solving two problems above is proposed, and the numerical examples show its feasibility.
Throughout, let denote the set of all complex matrices, the set of all real matrices, and the set of all matrices in with rank . The symbols , , , , , and , respectively, stand for the identity matrix with the appropriate size, the conjugate, the transpose, the conjugate transpose, the Moore-Penrose inverse, and the Frobenius norm of . For , ; represents the Hadamard product of and .
A linear model of image restoration is a matrix-vector equation is where represents the observed image, the original true image, additive noise, and a blurring matrix. Image restoration is to minimize blur in an observed image, namely, recover a optimal approximation of by given and , and get some statistical information of . The process of the image restoration for the model (1) can be described as where is a small positive parameter. It is known that is the least squares solution of (1) with minimal norm. However, for , the solution to (2), that is, in (1) is not feasible. We know if , then the solution converges to . In order to obtain a solution sufficiently near to , we usually take small such that , where the error norm is given.
Now we consider the generalized problem of the process of the image restoration.
Problem 1. Given , , , , being positive integers, , , , and , . Let
Find such that
The constraint Procrustes problem associated with several kinds of sets , that is, and in (3) has been extensively studied, such as the orthogonal Procrustes problem with being the set of orthogonal matrices , the symmetric Procrustes problem , -symmetric Procrustes problem , Hermitian, Hermitian -symmetric and Hermitian -skew-symmetric Procrustes problems , the Procrustes problems with constrained to the cone of symmetric positive semidefinite and symmetric elementwise matrices , and the generalized Procrustes analysis . The optimal approximation problem (4) is initially proposed in the processes of testing or revising given data. A preliminary estimate of the unknown matrix in can be obtained from experimental observation values and the information of statistical distribution.
We characterize the case , , , in Problem 1 and describe it as follows.
Problem 2. Given , , , and being positive integers, , , , . Let When is nonempty, find such that
Motivated by the work mentioned above, in this paper we mainly discuss the above two problems associated with being the set of Hermitian -conjugate matrices.
Recall that an complex matrix is -conjugate if , where is a nontrivial involution, that is, , , which was defined in . A matrix is Hermitian -conjugate if and , where . The Hermitian -conjugate matrix is very useful in scientific computation and digital signal and image processing, its special case, for example, Hermitian Toeplitz matrix, have been studied by several authors, see [14, 16–21]. We denote the set of all Hermitian -conjugate matrices by . Let , , and denote the set of all complex -conjugate matrices, real symmetric matrices, real skew-symmetric matrices, respectively.
This paper is organized as follows. In Section 2, we give some preliminary lemmas. In Section 3, we derive the expression of the unique solution to the Problem 1 with . In Section 4, we establish the solvability conditions for existence and an expression of the solution for Problem 2 with . In Section 5, we give examples to illustrate the results obtained in this paper.
For a nontrivial symmetric involutory matrix , there exist positive integers and such that and an orthogonal matrix such that where and . The columns of form an orthogonal basis for the eigenspace of associated with the eigenvalue .
Throughout this paper, we always suppose the nontrivial symmetric involutory matrix is fixed which is given by (7).
Lemma 4. For any matrix , , where and denotes the direct sum.
Proof. For , it is obvious that , where are are defined in (9). Hence, we just need to prove .
For any matrix , it is obvious that , where and are defined in (9).
We prove the uniqueness of . Note that that is, . If there exist and satisfying such that , then Multiplying on the left and right side and then taking the conjugate for (12), it yields implying and .
So holds, where , , and are defined in (9).
Proof. (i) It yields from (7) and (8) that
implying , that is, .
Conversely, if , according to the proof of the necessity, we can get .
The proof of (ii) can be analogously completed according to the proof of (i).
3. The Solution to Problem 1 with
We, in this section, give the explicit expression of the solution to Problem 1 with . In the following, we refer to the in .
According to Lemma 3, if , then where is defined as (8) and . Let By Lemma 5, it is easy to verify , . We obtain It follows from the unitary invariance of Frobenius norm, (21), (22), and that Suppose then
We first give the following lemma which can be obtained by contrast with Lemma 2.1 in .
Lemma 6. Given ; therefore, the singular value decomposition (SVD) of can be described as where Let , , , , and with , ; . Then is consistent if and only if where is arbitrary.
Theorem 7. Given , , and positive integers , , , and , where , , the notations , , , , , , , , , are defined as (18), (19), (20), (24), and (25), respectively. Let the SVD of be of the form (27) with (28). Then
Theorem 8. Given , the equation (4) is consistent if and only if
Proof. Obviously, is a closed convex set. Hence, there exists the unique element such that (4) holds. By applying Theorem 7 and the unitary invariance of Frobenius norm, for , we get Then is equivalent to (34) holds if and only if Substituting (35) into , we get is (32).
Corollary 9. if and only if
4. The Solution to Problem 2 with
We refer to in in the following text. In this section, we establish necessary and sufficient conditions for the existence and the expression of . When is nonempty, we present the expression of the unique solution to (6).
It follows from (21) that the system , , with unknown is consistent if and only if there exists such that . We first give the following lemma.
Lemma 11 (see Theorem 1 in ). Given . Let the SVD of be (27) with (28). Then the matrix equation has a symmetric solution if and only if and the symmetric solution can be expressed as where is arbitrary.
Theorem 12. Given , , , and positive integers and . The notations , , , , , , , , , are defined as (18), (19), (20), (24), and (25), respectively. Let the SVD of be of the form (27) with (28). Then the solution set is nonempty if and only if (37) holds, in which case,
Proof. If the solution set is nonempty, then there exists a matrix such that , , . We know that , , with is consistent if and only if there exists such that . By Lemma 11, is solvable for if and only if (37) holds, and the expression of the solution is (38). Insert (38) into (17), then we obtain is of the form (39).
Conversely, assume (37) holds, according to the proof of the necessity, , , is solvable for . For , by Lemma 3, .
Theorem 13. Given and is nonempty. Let . Then (6) is consistent for if and only if In particular, if , then
Proof. By Lemma 4, , where When is nonempty, for , we get Since , , by Lemma 3, we obtain , . Note that , then Hence, is equivalent to For the orthogonal matrix , we get Then is solvable if and only if It follows from (27) that By (48) and (47) we get Hence, from (39) and (49), we obtain which can be expressed as (40).
5. Numerical Examples
Algorithm 15. Input , , , ;
compute , , , , , , , , , by (18), (19), (20), (24), and (25);
make the SVD of with the form of (27) and compute , , , by (28);
if (37) holds, continue, or go to step 6;
input , compute the solution to Problem 2 with by (40);
input , compute the solution of Problem 1 with by (32).
Example 2. Let , be unitary matrices,and , where and is a zeros matrix. Let and ,Then compute for . Obviously, Problem 2 with is consistent for each value of . For matrices , obtained above, we first use Algorithm 15 to obtain the Hermitian -conjugate solutions approximate to , then compute the solutions of by MATLAB procedure . Let denote the solutions computed by Algorithm 15 and the solutions by MATLAB procedure . Let
Analysis of Results. As a general observation from Table 1, we find that the performance of Algorithm 15 to solve Problem 2 is very good and that of the MATLAB procedure is quite sensitive to the conditioning of matrix . For , both methods behave well. In this case we should choose MATLAB procedure to solve Problem 2 with for it is simple. However, as some singular values of are close to zero, the solutions computed by MATLAB procedure do not satisfy and gradually lose the property of Hermitian -conjugate matrix, while Algorithm 15 does it well. Hence, when has small singular values close to zero, the Algorithm 15 predominates over MATLAB procedure .
In this paper, we converted the Hermitian -conjugate generalized Procrustes problem to real symmetric Procrustes problem trickly and obtained its solution set. We also investigated the Hermitian -conjugate solution to the linear system of complex matrix equations , , and established solvable conditions and the formula for the Hermitian -conjugate solution. Moreover, we showed the optimal approximation solution to a given matrix in the above two corresponding solution set is unique, respectively. As applications, a numerical algorithm has been given and the examples have illustrated the feasibility of the algorithm. Additionally, we can further consider the least square Hermitian -conjugate solution of the system , ( is positive integer) and the corresponding optimal approximation problem.
The authors are very much indebted to the anonymous referees for their constructive and valuable comments and suggestions which greatly improved the original paper. This research was supported by the Innovation Foundation of Shanghai Municipal Education Commission (13ZZ080), the National Natural Science Foundation of China (11391240185), the Shanghai University Scientific Selection and Cultivation for Outstanding Young Teachers in special fund (sjr10009), and the Natural Science Foundation of Guangxi Province (no. 2012GXNSFBA053006).
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