Reconstruction of the Hermitian matrix by its spectrum and spectra of some number of its perturbations

Abstract

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 N², 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.