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

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.


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.

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.

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 ∈ 1 × , first of all, we have the following conclusion from Definition 1.
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 ∈ 1 × , we can derive from (1) and (3) the following: = − , + = 0 can be obtained, and, relying on Conclusion 1, ∈ 1 × can be obtained.
There will be The equations = − and + = are provided. Finally the proof of Theorems 3 and 4 is completed. Algebra 3 From Theorems 3 and 4 we have a corollary as follows.

The Unique Solutions on the First-Class Special Symmetric Matrix for Problem 2
Consider the following theorem.
The proof is completed.

Conditions for the Existence and Expression of General Solutions Based on the Second-Class Special Symmetric Matrix for Problem 1
Consider the following theorem.
can be obtained and ∈ 2 × will be provided. When = 2 , in the same way, we can prove the above theorem.
The proof is completed.

Theorem 8.
When , ∈ × , ∈ 2 × , and and ( = 1, 2) are given by (16), the sufficient and necessary conditions for the existence of solution = are And the general solution is as follows: Meanwhile 1 ∈ ( − )× , 2 ∈ ×( − ) , and according to (17) we can obtain singular value decompositions: Proof. Relying on Theorem 7, if there is ∈ 2 × , it will have Because is an orthogonal matrix and = , We can derive from (16), (48), and (49) It will be known that, when ∈ × , ∈ × , and rank( ) = [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.

The Unique Solutions on the Second-Class Special Symmetric Matrix for Problem 2
Consider the following theorem.
The proof is completed.