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A class of constrained inverse eigenproblem and associated approximation problem for skew symmetric and centrosymmetric matrices. (English) Zbl 1080.15013

The paper deals with a class of constrained inverse eigenproblems and associated approximation problems.
A real matrix \(A\), of size \(n \times n\), is called skew symmetric and centrosymmetric if \(A^T=-A\) and \(JAJ=A\), where \(J\) denotes the \(n \times n\) “backward identity” matrix. The set of these matrices is denoted by \(ACSR^{n \times n}\).
The authors study the following problems:
Problem I. Let \(b>0\) a real number, \(Z\) an \(n \times k\), \(k<n\), complex matrix of rank \(k\) and \(D_0=\text{diag}(\lambda_1i,\ldots,\lambda_ki)\) with \(\lambda_j\) a real number, \(j=1,2,\ldots,k\). Find \((Z,D_0)\) such that the set \(\varphi(Z,D_0)= \{A \in ACSR^{n \times n} \mid AZ=ZD_0\}\) is nonempty and find the subset \(\varphi(Z,D_0,b) \subset \varphi(Z,D_0)\) such that the imaginary parts of all of the remaining eigenvalues of any matrix in \(\varphi(Z,D_0,b)\) are located in the interval \([-b,b]\).
Problem II. Given and \(n \times n\) real matrix \(B\), find \(A_B \in \varphi(Z,D_0,b)\) such that \[ \| B-A_B\| = \min_{A \in \varphi(Z,D_0,b)} \| B-A\| , \] where \(\| \cdot \| \) is the Frobenius norm.
The authors show that the mentioned problems are essentially decomposed into the same kind subproblems for real antisymmetric matrices with smaller dimensions. They present the general solution of Problems I and II in the real number field. They also obtain the explicit expression of the nearest matrix to a given matrix in the Frobenius norm.

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15A29 Inverse problems in linear algebra
15B57 Hermitian, skew-Hermitian, and related matrices
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References:

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