The Centrosymmetric Matrices of Constrained Inverse Eigenproblem and Optimal Approximation Problem

In this paper, a kind of constrained inverse eigenproblem and optimal approximation problem for centrosymmetric matrices are considered. Necessary and sufficient conditions of the solvability for the constrained inverse eigenproblem of centrosymmetricmatrices in real number field are derived. A general representation of the solution is presented for a solvable case. (e explicit expression of the optimal approximation problem is provided. Finally, a numerical example is given to illustrate the effectiveness of the method.


Introduction
Inverse eigenproblems arise in a remarkable variety of applications, including control theory [1,2], vibration theory [3,4], structural design [5], molecular spectroscopy [6], in developing numerical methods, and the ordinary and partial differential equation solving [7,8]. Centrosymmetric matrices are applied in information theory, linear system theory, and numerical analysis theory [9]. e unconstrained centrosymmetric matrices' problems have been discussed [9][10][11][12][13][14], a class of unconstrained matrices' inverse eigenproblems has been obtained [15][16][17][18], and the constrained inverse eigenproblems have been discussed [19][20][21][22], but only when the eigenvalues are real or imaginary numbers. For general real matrices, the eigenvalues are not necessarily real or imaginary numbers, so when the eigenvalue is complex, it is difficult to find the constraint solution. In this paper, we will use the real Schur decomposition theorem and the similar decomposition theorem and introduce a new norm to get the corresponding expression of the best approximation solution.
roughout the paper, we denote the set of real n × m matrices, real n × n orthogonal matrices, n × n centrosymmetric matrices, and real numbers, respectively, by R n×m , OR n×n , CSR n×n , and R. A T , A + , rank(A), and ‖A‖ F denote the transpose, the Moore-Penrose generalized inverse, the rank, and the Frobenius norm of a matrix A, respectively. I is the identity matrix. λ(A) denotes the set of eigenvalues of the matrix A. λ(A)/λ(B) denotes the set of the difference of λ(A) and λ(B). D r is a closed disc with radius r and center origin, [A] D r denotes the square matrix A with all of its eigenvalues located in the closed disc D r , and R n×n D r denotes the set of n × n matrices with their eigenvalues located in the disc D r . e notation A 11 ⊕ A 22 ⊕ · · · ⊕ A kk denotes the direct sum of the matrices A 11 , A 22 , . . . , A kk , where A jj ∈ R n j ×n j . Let e i be the ith standard unit vector, and the matrix S n � (e n , e n− 1 , . . . , e 1 ). [x] represents the largest integer less than or equal to x. e following definition is given in [23].
Clearly, each eigenvalue of matrix A ∈ CSR n×n is either a real number or a complex number, if A has complex eigenvalues, and they must occur in complex conjugate pairs, . . , k) are eigenvectors associated with λ j and λ j , respectively; also, we have If A has real eigenvalues, then λ j ∈ R(j � k + 1, . . . , l) are eigenvalues of A and x j ∈ R n (j � k + 1, . . . , l) are eigenvectors associated with λ j ; also, we have where let X j � x j , en, 2k + (l − k) � n.

Remark 1.
Here, r-multiple eigenvalues are counted r-times and their eigenvectors may be linearly dependent.

Definition 2.
Let A ∈ R n×n , given Y ∈ R n×n , and rank(Y) � n. en, we define a new norm called the Y norm of matrix A as In contrast to the definition of matrix norm (see [24], Definition 5.1.1), it is easy to show that ‖·‖ Y is a kind of matrix norm. Now, we can present the optimal approximation of constrained inverse eigenproblem of centrosymmetric matrices as follows: Constrained Inverse Eigenproblem. Given a real number r > 0. Let X � (X 1 , X 2 , . . . , X l ) ∈ R n×m , Λ � diag(Λ 1 , Λ 2 , . . . , Λ l ) ∈ R m×m , l ≤ m ≤ n, where X j and Λ j satisfy equation (2) or equation (4). D r � Z | |Z| ≤ r, Z ∈ C { } is a given closed disc. Find matrix A such that the set is nonempty, and find the subset S D r ⊂ S such that the remaining eigenvalues of any matrix in S are located in the disc D r . Optimal Approximation Problem. Given A * ∈ R n×n , ‖ ·‖ Y is the norm which has been defined as equation (5), Y is an invertible matrix concerning with A * and A ∈ S D r ; find a matrix A ∈ S D r , such that where S D r is the solution set of the Constrained Inverse Eigenproblem.
is paper is organized as follows. In Section 2, we provide the solvability conditions of the Constrained Inverse Eigenproblem and its general solution in that case. In Section 3, we get the expression of the solution for Optimal Approximation Problem. In Section 4, we give an algorithm of Constrained Inverse Eigenproblem and Optimal Approximation Problem and give a numerical example of Optimal Approximation Problem.

The Solvability Conditions and General Solution of Constrained Inverse Eigenproblem
Firstly, let k � [n/2] and characterize the set of all centrosymmetric matrices as follows.
When n � 2k, let When n � 2k + 1, let Clearly, D is an orthogonal matrix for all of the n. Secondly, given X ∈ R n×m , denote Decomposing the matrices X, X D 1 , and X D 2 by the SVD, we have where where

Lemma 1 (see [9], Lemma 2).A ∈ CSR n×n if and only if A can be expressed as
where, if n � 2k, then D is the form of equation (8), and if n � 2k + 1, then D is the form of (9).

Mathematical Problems in Engineering
Lemma 2 (see [15], eorems 7.2 and 7.3). Given X ∈ R n×m , Λ ∈ R m×m , and X decomposed as equation (11), let where If the condition is satisfied, the general solution can be expressed as and where X D 1 , X D 2 , V 2 , and Q 2 is the same as equations (10)- (14). Let where B ij � U T i BU j and C ij � P T i CP j (i, j � 1, 2), and the set of the general solution can be expressed as where Proof. If S is nonempty, by Lemma 1, we have Using equation (10), equation AX � XΛ is equivalent to It follows from Lemma 2, and we can obtain AX � XΛ is solvable if and only if equation (19) holds, and the solution set can be expressed as equation (21)

Theorem 1. Constrained Inverse Eigenproblem is solvable if and only if
e general solution can be expressed as where

Lemma 5. Given a matrix
By similarity invariant, we can derive an invertible matrix

there exists a unique matrix:
such that Proof. From we may obtain Mathematical Problems in Engineering 3 Let (d j /c j ) � tan θ j , (b j /a j ) � tan θ, 0 ≤ θ, θ j ≤ π, s j > r, then equation (32) is equivalent to It is easy to see that f(θ, r) � min if and only if θ � θ j and r � s j .
Note [B jj ] D r � A jj , from equation (33), we may show that equation (29) implies that ere, we note [B jj ] D r � A jj just for the sake of writing. To show eorem 2, we first introduce some notations as follows: where  (35), the orthogonal matrices U and P are given in equations (13) and (14), respectively; then, there exist orthogonal matrices Q * 1 and Q * 2 and inverse matrices Y 1 and Y 2 , such that where Λ jj � l j f j − f j l j , j � 1, 2, . . . , s.
Proof. For Lemma 7, there exist orthogonal matrices Q * 1 ∈ OR (n− k− r 1 )×(n− k− r 1 ) and Q * 2 ∈ OR (k− r 2 )×(k− r 2 ) such that where (39) □ For Lemma 5, there exist inverse matrices Y 1j and Y 2j , satisfying Then, Optimal Approximation Problem has a unique solution which can be expressed as where Proof. Let A ∈ S D r , by Definition 2, we have e expression of A is the same as equation (25) and D T A * D is decomposed as equation (35); then, equation (44) is equivalent to Obviously, we may derive that equation (44) has the solution A if and only if From Lemmas 5 and 6, equations (36) and (37), we may see that equations (46) and (47) imply that Mathematical Problems in Engineering 5 where (51) where By equations (48) and (49), we have Note Equations (55), (56), and (25) imply equation (42). □

Numerical Example
Based on eorems 1 and 2, we propose the following algorithm for solving Constrained Inverse Eigenproblem and Optimal Approximation Problem (Algorithm 1).
In this section, we will give a numerical example to illustrate our results. All the tests are performed by MATLAB6.5.
The matrices X, Λ, and A * and the radius r are given by following: Step 1. Input X, Λ, A * , r; Step 2. Decompose D T X as equation (10); Step 3. Calculate the singular value decomposition X D 1 , X D 2 as equations (13) and (14), verify that equation (24) is true and A can be expressed as equation (25); Step 4. Decompose D T A * D as equation (35) (50) and (52), respectively; Step 7. Calculate A by eorem 2.
ALGORITHM 1: e solution of (7). 6 Mathematical Problems in Engineering By algorithm, the constrained optimal approximation solution A by Optimal Approximation Problem is In the practical engineering problems, it is usually required that the matrix is a centrosymmetric matrix and the partial eigenvalues of the matrix must be located in a given closed disc or interval. We verify that A in Optimal Approximation Problem is a centrosymmetric matrix and the eigenvalues of λ(A)/λ(Λ) are lost in the disc D r .

Data Availability
All data generated or analyzed during this study are included in this article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.