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The solvability conditions for the inverse eigenvalue problems of centro-symmetric matrices. (English) Zbl 1028.15012

A matrix \((a_{ij})\in \mathbb{R}^{n\times n}\) is called centro-symmetric if \(a_{ij}=a_{n+1-i,n+1-j}\). The following problem is solved in this paper.
Given \(X\in \mathbb{R}^{n\times m}\), \(\Lambda=\text{diag}(\lambda_1,\dots,\lambda_{m}) \in \mathbb{R}^{m\times m}\). Find (all) centro-symmetric matrices \(A\) such that \(AX=X\Lambda\).
A necessary and sufficient condition for the existence of such an \(A\) is obtained and set \(S_E\) of all solutions is described. Moreover for \(\tilde A\in \mathbb{R}^{n\times n}\) an expression for the unique \(A^\ast\in S_{E}\) for which \(\|\tilde A-A^\ast\|= \inf_{A\in S_{E}}\|\tilde A-A\|\) is written (here \(\|B\|=\text{tr}(B^{\top}B)\) for any \(B\in \mathbb{R}^{n\times n}\)).

MSC:

15A29 Inverse problems in linear algebra
15A18 Eigenvalues, singular values, and eigenvectors
65F18 Numerical solutions to inverse eigenvalue problems
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