## 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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### References:

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