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Reconstruction of the Hermitian matrix by its spectrum and spectra of some number of its perturbations. (English. Russian original) Zbl 1359.15009

Dokl. Math. 94, No. 2, 529-531 (2016); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 470, No. 3, 257-259 (2016).
Summary: Explicit formulas for matrix elements of the Hermitian matrix are found through a spectrum of this matrix and spectra of some number of its perturbations. A dependence of sufficient number of perturbations from the structure of the matrix and the kind of perturbations is established. It is shown that for arbitrary matrix needed number of perturbations is of \(N^2\), where \(N\) is an order of the matrix. In the case, when the number and locations of zero elements of the matrix is known, needed number of perturbations decreases essentially.

MSC:

15A29 Inverse problems in linear algebra
15A18 Eigenvalues, singular values, and eigenvectors
15B57 Hermitian, skew-Hermitian, and related matrices
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References:

[1] Ambarzumian, V. A., No article title, Z. Phys., 53, 690-695 (1929) · JFM 55.0868.01 · doi:10.1007/BF01330827
[2] Borg, G., No article title, Acta Math., 78, 1-96 (1946) · Zbl 0063.00523 · doi:10.1007/BF02421600
[3] V. A. Marchenko and V. V. Slavin, Inverse Problems in the Theory of Small Oscillations (Naukova Dumka, Kiev, 2015) [in Russian].
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