Algebra

Volume 2014 (2014), Article ID 392467, 7 pages

http://dx.doi.org/10.1155/2014/392467

Research Article

A Study of Inverse Problems Based on Two Kinds of Special Matrix Equations in Euclidean Space

Rui Huang,¹ Xiaodong Wu,¹ Ruihe Wang,² and Hui Li³

¹College of Petroleum Engineering, China University of Petroleum, Beijing 102249, China

²China National Oil and Gas Exploration and Development Corporation, Beijing 100034, China

³CNPC Beijing Richfit Information Technology Co., Ltd., Beijing 100013, China

Received 11 January 2014; Accepted 13 April 2014; Published 26 May 2014

Academic Editor: Ricardo L. Soto Montero

Copyright © 2014 Rui Huang 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.

Abstract.

Two special classes of symmetric coefficient matrices were defined based on characteristics matrix; meanwhile, the expressions of the solution to inverse problems are given and the conditions for the solvability of these problems are studied relying on researching. Finally, the optimal approximation solution of these problems is provided.

1. Introduction

In recent years, a lot of matrix problems have been used widely in the fields of structural design, automatic control, physical, electrical, nonlinear programming and numerical calculation, for example, a matrix Eigen value problem was applied for mixed convection stability analysis in the Darcy media by Serebriiskii et al. [1] and some of the problems based on the nonskew symmetric orthogonal matrices were studied by Hamed and Bennacer in 2008 [2], but some of the matrix inverse problems still need further research in order to make it easier to discuss relevant issues. Therefore, in this paper, we studied the inverse problems of two kinds of special matrix equations based on the existing research achievements, moreover, the expressions and conditions of the matrix solutions are given by related matrix-calculation methods. Some definitions and assumptions of the inverse problem for two forms of special matrices are given in Section 2. In Sections 3 and 5 we discuss the existence and expressions of general solution based on the two classes of matrices, and in Sections 4 and 6 we prove the uniqueness of matrices for researching related inverse problems.

2. Definitions and Assumptions of Inverse Problems for Two Forms of Special Matrices

In order to research some inverse problems of related matrices, we give the following definitions and assumptions.

Definition 1. When , , , , , and , will be called the first-class special symmetric matrix and the set of these special symmetric matrices is denoted by . The corresponding problems are as follows.

Problem 1. When , can be obtained, so that .

Problem 2. When , can be obtained, so that , where is the solution set of the first problem.

Definition 2. When , and , will be called the second-class special symmetric matrix and the set of these special symmetric matrices is denoted by . The corresponding problems are as follows.

Problem 1. When , can be found, so that .

Problem 2. When , can be found, so that , where is the solution set of the first problem.

3. Existence and Expression of General Solutions Based on the First-Class Special Symmetric Matrix for Problem 1

To research the structure and properties of the special symmetric matrix , first of all, we have the following conclusion from Definition 1.

Conclusion 1. Consider , and the sufficient and necessary conditions for are

Theorem 3. Consider  , and the sufficient and necessary conditions for are

where , , and is an orthogonal matrix. Consider and .

Proof. Rely on the decomposition theorem of symmetric orthogonal matrix [3, 4]. When is a symmetric orthogonal matrix, , the can be represented as the following equation by an orthogonal matrix ,  :

where is identity matrix [5].

When , we can derive from (1) and (3) the following:

Based on and , there will be

and can be derived from (4) and (5).

Conversely, when

, can be obtained, and, relying on Conclusion 1,   can be obtained.

Theorem 4. When and the singular value decomposition is

where , , and , , , , , , is the set of orthogonal matrix, and is Moore-Penrose generalized inverse matrix [6–8].

The sufficient and necessary conditions for the existence of solution are

And the general solution is as follows:

where .

Proof. Define

where and ,  .

The equation can be represented as the following equation by (7) and (10):

According to (7) and (8), there will be

Then (11) is equivalent to the following equation:

and and can be obtained from (12) and (13), so the equation has solutions, . And the general solution can be represented as the following equation:

where .

Conversely, when the equation has solutions and , and can be obtained relying on the Penrose theorem [9, 10].

There will be

The equations and are provided.

Finally the proof of Theorems 3 and 4 is completed.

From Theorems 3 and 4 we have a corollary as follows.

Corollary 5. Consider that  , and , , and are given by these equations:

The singular value decompositions are

where , , , , , , , , , , , .

The sufficient and necessary conditions for the existence of solution are

And the general solution is as follows:

where and .

The proof is completed.

4. The Unique Solutions on the First-Class Special Symmetric Matrix for Problem 2

Consider the following theorem.

Theorem 6. When , , and and are given by (16) and obey (18), Problem 2 has the unique solution .

Define

Then the unique solution can be represented as follows:

where

is defined by the following equation:

Finally can represent the following set by Theorem 4:

Consider and .

Proof. Define these following equations:

where and .

Because of we can obtain

Finally,

When , and , we can obtain:

When and , so that ,

can be derived from (26) and

can be obtained according to (17), (18), and (24).

Therefore,

Relying on (27) the solution can be provided as follows:

where

and can be obtained because of and .

Finally, rely on the optimal approximation [11, 12], and is a closed convex set; Problem 2 has the unique solution , , so that .

The proof is completed.

5. Conditions for the Existence and Expression of General Solutions Based on the Second-Class Special Symmetric Matrix for Problem 1

Consider the following theorem.

Theorem 7. Consider that , and the sufficient and necessary conditions for are

where is an orthogonal matrix. and .

When , define

When , define

, .

Proof. First relying on the definition and property of the matrix , can represent the following set.

 When ,

   and  . When ,

 .

We first discuss the topic .

From (37) and (39),

Define , .

According to and , we have and , and

can be obtained from (42).

Conversely, when , , , can be obtained and will be provided.

When , in the same way, we can prove the above theorem.

The proof is completed.

Theorem 8. When , , and and    are given by (16), the sufficient and necessary conditions for the existence of solution are

And the general solution is as follows:

where

Meanwhile , , and according to (17) we can obtain singular value decompositions:

Proof. Relying on Theorem 7, if there is , it will have

Because is an orthogonal matrix and ,

We can derive from (16), (48), and (49)

It will be known that, when , , and [13], the sufficient and necessary conditions for the existence of solution are and the general solution is

There will be

The general solution can obtain

The sufficient and necessary conditions for the existence of solution are .

The proof is completed.

6. The Unique Solutions on the Second-Class Special Symmetric Matrix for Problem 2

Consider the following theorem.

Theorem 9. When ,  , and and    are given by (16) and (17) and obey (18), Problem 2 has the unique solution .

The unique solution can be represented as follows:

where

  , :

  , :

From Theorem 9   can be represented as follows:

Proof. Defining this following equation:

where , ,

, :

For the following equation can be obtained:

When is known and relies on the definition of , there will be the unique set and to make the following equations true:

From (60), (63), and (65) we obtain

Equation (67) can be known from (17) and (64) as follows:

Therefore, is equivalent to

and , can be obtained.

From the above results and (63), the solution ) can be represented as follows:

Finally, because is a closed convex set, Problem 2 has the unique solution , , so that .

The proof is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are grateful to the referee for the useful comments and valuable suggestions.

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