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.
Highlights
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
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
Summary
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]. Λ(A) denotes the set of eigenvalues of the matrix A. Where SDr is the solution set of the Constrained Inverse Eigenproblem. 2. The Solvability Conditions and General Solution of Constrained Inverse Eigenproblem. Let k [n/2] and characterize the set of all centrosymmetric matrices as follows. Given X ∈ Rn×m, Λ ∈ Rm×m, and X decomposed as equation (11), let. 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) .
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