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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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