Abstract
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 centrosymmetric matrices in real number field are derived. A general representation of the solution is presented for a solvable case. The explicit expression of the optimal approximation problem is provided. Finally, a numerical example is given to illustrate the effectiveness of the method.
1. 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]. The unconstrained centrosymmetric matrices’ problems have been discussed [9–14], a class of unconstrained matrices’ inverse eigenproblems has been obtained [15–18], and the constrained inverse eigenproblems have been discussed [19–22], 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.
Throughout the paper, we denote the set of real matrices, real orthogonal matrices, centrosymmetric matrices, and real numbers, respectively, by , , , and R. , , , and denote the transpose, the Moore–Penrose generalized inverse, the rank, and the Frobenius norm of a matrix A, respectively. I is the identity matrix. denotes the set of eigenvalues of the matrix A. denotes the set of the difference of and . is a closed disc with radius r and center origin, denotes the square matrix A with all of its eigenvalues located in the closed disc , and denotes the set of matrices with their eigenvalues located in the disc . The notation denotes the direct sum of the matrices , where Let be the standard unit vector, and the matrix . represents the largest integer less than or equal to x. The following definition is given in [23].
Definition 1. is called centrosymmetric matrix if , .
Clearly, each eigenvalue of matrix is either a real number or a complex number, if A has complex eigenvalues, and they must occur in complex conjugate pairs, note that , are eigenvalues of A, and are eigenvectors associated with and , respectively; also, we havewhere letIf A has real eigenvalues, then are eigenvalues of A and are eigenvectors associated with ; also, we havewhere letThen, .
Remark 1. Here, r-multiple eigenvalues are counted r-times and their eigenvectors may be linearly dependent.
Definition 2. Let , given and . Then, we define a new norm called the Y norm of matrix A asIn contrast to the definition of matrix norm (see [24], Definition 5.1.1), it is easy to show that 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 . Let where satisfy equation (2) or equation (4). is a given closed disc. Find matrix A such that the set is nonempty, and find the subset such that the remaining eigenvalues of any matrix in S are located in the disc . Optimal Approximation Problem. Given , is the norm which has been defined as equation (5), Y is an invertible matrix concerning with and ; find a matrix , such thatwhere is the solution set of the Constrained Inverse Eigenproblem.This 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.
2. The Solvability Conditions and General Solution of Constrained Inverse Eigenproblem
Firstly, let and characterize the set of all centrosymmetric matrices as follows.
When , let
When , letClearly, D is an orthogonal matrix for all of the n.
Secondly, given , denote
Decomposing the matrices , and by the SVD, we havewhere , andwhere and where and
Lemma 1. (see [9], Lemma 2). if and only if A can be expressed aswhere, if , then D is the form of equation (8), and if , then D is the form of (9).
Lemma 2. (see [15], Theorems 7.2 and 7.3). Given , and X decomposed as equation (11), letwhere . Equation is consistent if and only ifIf the condition is satisfied, the general solution can be expressed asand is arbitrary.
Lemma 3. Given . Equation is consistent with if and only ifwhere is the same as equations (10)–(14). Letwhere , and the set of the general solution can be expressed aswhere are arbitrary.
Proof. If S is nonempty, by Lemma 1, we haveUsing equation (10), equation is equivalent toIt follows from Lemma 2, and we can obtain is solvable if and only if equation (19) holds, and the solution set can be expressed as equation (21) easily.
Remark 2. Equation (21) implies that if and only if arbitrary matrices and . From Lemmas 1–3, we have the following theorem.
Theorem 1. Constrained Inverse Eigenproblem is solvable if and only if
The general solution can be expressed aswhere , , , and are arbitrary matrices.
3. The Solution of Optimal Approximation Problem
Lemma 5. Given a matrixBy similarity invariant, we can derive an invertible matrix and such thatwhere and .
Lemma 6. Given then, there exists a unique matrix:such that
Proof. Fromwe may obtainLet then equation (32) is equivalent toIt is easy to see that if and only if .
Note , from equation (33), we may show that equation (29) implies thatThere, we note just for the sake of writing.
To show Theorem 2, we first introduce some notations as follows:where
Lemma 7. (see [16], Theorem 7.6). If then there exists an orthogonal matrix , such that is upper quasi-triangular, that is, is block upper-triangular where each diagonal block is either a matrix or a matrix having complex conjugate eigenvalues.
Lemma 8. Given matrix , is decomposed as equation (35), the orthogonal matrices U and P are given in equations (13) and (14), respectively; then, there exist orthogonal matrices and inverse matrices and , such thatwhere where
Proof. For Lemma 7, there exist orthogonal matrices such thatwhere
For Lemma 5, there exist inverse matrices and , satisfying
Let and , and are identity matrices. Then, it is easy to see that equations (36) and (37) exist.
Theorem 2. Given , suppose given matrices real number make Constrained Inverse Eigenproblem solvable, and the solution expression is the same as equation (25). LetThen, Optimal Approximation Problem has a unique solution which can be expressed aswhere
Proof. Let by Definition 2, we haveThe expression of A is the same as equation (25) and is decomposed as equation (35); then, equation (44) is equivalent toObviously, we may derive that equation (44) has the solution if and only ifFrom Lemmas 5 and 6, equations (36) and (37), we may see that equations (46) and (47) imply thatwherewhereBy equations (48) and (49), we haveNoteEquations (55), (56), and (25) imply equation (42).
4. Numerical Example
Based on Theorems 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, , and and the radius r are given by following:
By algorithm, the constrained optimal approximation solution 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 in Optimal Approximation Problem is a centrosymmetric matrix and the eigenvalues of are lost in the disc .
Data Availability
All data generated or analyzed during this study are included in this article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Research supported by the Scientific Research of Hunan Provincial Education Department (Grant no. 18B419) and Science and Technology Project of Changsha (Grant no. kc1809026).