Journal of Applied Mathematics

Journal of Applied Mathematics / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 696019 | https://doi.org/10.1155/2013/696019

Juan Yang, Yuan-bei Deng, "Procrustes Problems for General, Triangular, and Symmetric Toeplitz Matrices", Journal of Applied Mathematics, vol. 2013, Article ID 696019, 5 pages, 2013. https://doi.org/10.1155/2013/696019

Procrustes Problems for General, Triangular, and Symmetric Toeplitz Matrices

Academic Editor: Mohamad Alwash
Received15 Nov 2013
Accepted17 Dec 2013
Published30 Dec 2013

Abstract

The Toeplitz Procrustes problems are the least squares problems for the matrix equation over some Toeplitz matrix sets. In this paper the necessary and sufficient conditions are obtained about the existence and uniqueness for the solutions of the Toeplitz Procrustes problems when the unknown matrices are constrained to the general, the triangular, and the symmetric Toeplitz matrices, respectively. The algorithms are designed and the numerical examples show that these algorithms are feasible.

1. Introduction

Consider the constrained least-squares optimization problems: where , , and denotes the Frobenius norm

These problems have been investigated by a series of literatures. For instance, when , then is a solution (refer to [1]), while the symmetric matrix solution case and orthogonal matrix solution case are analyzed in [2] and [3] respectively. And the symmetric positive (semi-) definite least squares problem is discussed by Higham [4].

In this paper, the following Toeplitz least-squares problem are analyzed: where and are the set of Toeplitz matrices, and a matrix is called a Toeplitz matrix if its entries satisfy , for all , , and with , ,

In this work, we discuss problem (3) in details. In Section 2, the general Toeplitz problem is discussed, and in Section 3, the triangular Toeplitz problem and the symmetric Toeplitz problem are discussed, while in Section 4, we give algorithms and numerical examples.

We give some definition and lemmas first.

Definition 1 (see [5]). Matrix is called inverse matrix of when , and it can be denoted by or .

Lemma 2 (see [5]). Let ; if there exist nonsingular matrices and such that then the sufficient and necessary condition of is where ,, and .

Remark 3. When , , and , is denoted by .

Lemma 4 (see [6]). A finite dimensional linear subspace of a normed linear space contains at least one point of minimum distance from a fixed point.

Lemma 5 (see [7]). Let be a convex function continuously differentiable, and then any local minimum point is also a global minimum point. And is a solution of problem if and only if satisfies , where means the gradient of function .

Lemma 6 (see [7]). The quadratic function is convex if and only if is positive semidefinite.

2. The General Toeplitz Problem

We transform problem (3) into the following equivalent form: where , are defined as The objective function can be rewritten as The first term of the right side in (9) can be expressed as where Also the second term of the right side in (9) can be expressed as where , ,, .

It thus yields that

Taking partial derivative on about gives

By the first order necessary condition, we obtain Therefore the following theorem holds.

Theorem 7. The solution of problem (3) exists, and its general form can be expressed as where , for all means inverse of .

Proof. Let , where , and then the subspace is a normed linear space under Frobenius norm. It follows from Lemma 4 that there exists at least one matrix in such that . So, by the definition of , there exists at least one scalar constituting matrix , which means we can rewrite as , where . Therefore the solution of problem (3) exists.
It is apparent from (13) that the objective function is a quadratic function. Besides, by the following discussion, is also a convex function. Since is positive semi-definite, it follows that, for any , the expression of in (10) holds In other words, is a positive semidefinite matrix. Then according to Lemma 6, is a convex function. Since is continuously differentiable, we then get that any solution of (16) is also a solution of (7) by Lemma 5. Clearly, the solution of linear system of (16) is (see [5]). The required solution of problem (7) is then given by , where . The theorem follows by the equivalence of problem (3) and problem (7).

Theorem 8. Problem (3) has a unique solution if and only if has full rank. In this case, the unique solution is where .

Proof (necessity). If has full rank, then the linear system of (16) has a unique solution ; therefore the solution of problem (7) is determined uniquely, so is problem (3).
Sufficiency follows from the conversely procedure of necessity.

Corollary 9. If A has full rank in column, then the solution of problem (3) is unique.

3. The Triangular Toeplitz Problem and the Symmetric Toeplitz Problem

In this section, we discuss the triangular Toeplitz problem and symmetric Toeplitz problem.

3.1. The Triangular Toeplitz Problem

By the upper triangular Toeplitz problem, as the definition of [6], we mean the minimization problem where , are the subspace of upper triangular Toeplitz matrices, with elements of the general form

We transform problem (21) into the following problem where The proof procedure is similar to that in problem (7). In this case, the related function , the unknown , the linear system of equations, and the matrix as well as the scalar can be expressed as

Therefore we obtain a conclusion on upper triangular Toeplitz problem as follows.

Theorem 10. The solution of problem (21) exists, and its general form can be expressed as where , for all , means inverse of .

Theorem 11. Problem (21) has a unique solution if and only if has full rank. In this case, the unique solution is where .

Corollary 12. If A has full rank in column, then the solution of problem (21) is unique.

Similarly, we can solve the lower triangular Toeplitz problem.

3.2. The Symmetric Toeplitz Problem

The symmetric Toeplitz problem is the following minimization problem: where , are the subspace of symmetric Toeplitz matrices, with elements of the general form: We consider the following objective function: where

In this case, the related function , the unknown , the linear system of equations, and the matrix as well as the scalar can be expressed as

Therefore we obtain a conclusion about the symmetric Toeplitz problem.

Theorem 13. The solution of problem (28) exists, and its general form can be expressed as where , for all , and means inverse of .

Theorem 14. Problem (28) has a unique solution if and only if has full rank. In this case, the unique solution is where .

Corollary 15. If A has full rank in column, then the solution of problem (28) is unique.

4. Computation and Examples

The derivation of general Toeplitz problem leads to the following computational algorithm.

Algorithm 16. (1)Generate matrices according to the definition in (7).(2)Compute the matrix and scalar by formula (11) and (12).(3)If is of full rank, then solving the linear system of (16) directly, to obtain the solution ; else calculate , forming , for all . In general, let , and in Lemma 2 equal to , we can obtain one of inverse of , that is .(4)Compute from , where .

The algorithms for other special Toeplitz problems can be designed similarly.

We mention that, in the beginning of this paper, solutions for that two instances are actually obtained based on Algorithm 16. We can give another example of Toeplitz problem with matrix of full rank in column as follows. According to Corollaries 9, 12, and 15, the solution is unique.

Example 17. Consider the following
Using the algorithms mentioned above, computing the Toeplitz problems by Matlab, we obtain the following results.
The general Toeplitz matrix solution is The upper triangular Toeplitz matrix solution is and the symmetric Toeplitz matrix solution is

5. Conclusions

We have discussed the Toeplitz Procrustes problem including the general, triangular, and symmetric cases. After transforming the origin problem into a quadric form, we gain the general or unique solution with facility, by solving the linear system of equations.

Acknowledgments

The authors want to thank M. G. Eberle and M. C. Macle who have first proposed and studied the Toeplitz Procrustes problem. They also thank any referee for helpful suggestions or strengthening the theorems.

References

  1. G. H. Golub and C. F. van Loan, Matrix Computations, vol. 3, Johns Hopkins University Press, Baltimore, Md, USA, 1983. View at: MathSciNet
  2. N. J. Higham, “The symmetric Procrustes problem,” BIT. Numerical Mathematics, vol. 28, no. 1, pp. 133–143, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  3. N. J. Higham, “Computing the polar decomposition-with applications,” Society for Industrial and Applied Mathematics. Journal on Scientific and Statistical Computing, vol. 7, no. 4, pp. 1160–1174, 1986. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. N. J. Higham, “Computing a nearest symmetric positive semidefinite matrix,” Linear Algebra and its Applications, vol. 103, pp. 103–118, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. X. C. He and Y. W. Sun, Guangyi ni Juzhen Yinlun, Jiangsu Science and Technology Press, Nanjing, China, 1991. View at: MathSciNet
  6. E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill Book, New York, NY, USA, 1966. View at: MathSciNet
  7. L. D. Hui and T. X. Qiao, Numerical Optimization, Science press, Beijing, China, 2006.

Copyright © 2013 Juan Yang and Yuan-bei Deng. 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.


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