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The inverse eigenvalue problem for Hermitian anti-reflexive matrices and its approximation. (English) Zbl 1065.65057
A matrix $$A$$ is said to be anti-reflexive with respect to an orthogonal and symmetric matrix $$J$$ if $$A = -JAJ$$. Note that within the scope of inverse eigenvalue problems, it can be assumed without loss of generality that $J = \left[ \begin{matrix} I_r & 0 \\ 0 & -I_{n-r} \end{matrix} \right].$ In this case, $$A$$ is a Hermitian anti-reflexive matrix if and only if it takes the form $A = \left[ \begin{matrix} 0 & F \\ F^H & 0 \end{matrix} \right]$ for some $$r\times (n-r)$$ matrix $$F$$. This reveals a strong link between Hermitian anti-reflexive inverse eigenvalue problems and inverse singular value problems [cf. M. T. Chu, SIAM J. Numer. Anal. 29, No. 3, 885–903 (1992; Zbl 0757.65041)].
Given sets of vectors $$x_1,\dots,x_m$$ and scalars $$\lambda_1,\dots,\lambda_m$$, the author provides a complete characterization of $$S$$, the set of all Hermitian anti-reflexive matrices with eigenvector/eigenvalue pairs $$(x_i,\lambda_i)$$. Moreover, an explicit formula and a computational method for finding the matrix $$A^\star \in S$$ nearest to a given matrix $$A$$ are given. The paper is concluded by several numerical examples.

##### MSC:
 65F18 Numerical solutions to inverse eigenvalue problems
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