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The solvability conditions for the inverse eigenproblems of symmetric and generalized centro-symmetric matrices and their approximations. (English) Zbl 1109.65034

The authors investigate the inverse eigenvalue problem of constructing a matrix \(A\in S\subset \mathbb{R}^{n\times n}\) which satisfies \(AQ=Q\Lambda \) where \(Q\in \mathbb{R}^{n\times m}\) and \(\Lambda =diag\left( \lambda_{1},\dots, \lambda _{n}\right) \) are given matrices. They obtain conditions for the existence of a solution in the set \(S\) of symmetric and generalized centro-symmetric matrices defined by \(R_{2k}=\left[ \begin{matrix} E & FP \\ P^{T}F & P^{T}EP \end{matrix} \right]\) and \(R_{2k+1}=\left[ \begin{matrix} E & u & FP \\ u^{T} & \alpha & u^{T}P \\ P^{T}F & P^{T}u & P^{T}EP \end{matrix} \right]\) where \(E,F\in \mathbb{R}^{k\times k}\), \(P\) is an orthogonal matrix and \(\alpha \in \mathbb{R}\). They also examine the projection of a regular matrix onto \(S\) and give numerical examples at the end.

MSC:

65F18 Numerical solutions to inverse eigenvalue problems
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