Ein inverses Eigenwertproblem

AbstractThe paper ios concerned with the problem of finding a real, diagonal matrix M such that A + M has prescribed eigenvalues where A is any given symmetric matrix. This problem represents a discrete analog of the inverse eigenvalue problem in which we seek to determine a “potential” g(x) such that the operator in Hilbert space, l(y) = − y″ + g(x)y with appropriate boundary conditions, possesses a prescribed spectrum.Section 2 defines the “inverse problem” and its generalization.In Section 3, upper and lower bounds are given for the eigenvalues of a symmetric matrix, obtained by the iterative application of Temple's theorem. Also, an estimate is given for the changes in the eigenvalues produced by perturbing a given symmetric matrix by a diagonal matrix.Section 4 contains a complete discussion of the case of 2 × 2 matrices.In Section 5, from simple considerations concerning rotations in an n(n + 1)2-dimensional space, necessary conditions are given for the inverse problem to have a solution.In Section 6, the Brouwer fixed point theorem is applied to an operator equation which is equivalent to the “inverse problem.” A sufficient condition is given for the problem to have a solution: let k denote a certain seminorm on the set of all symmetric matrices and let d be the minimal mutual distance of the given eigenvalues. Then the inverse problem has a solution provided k(A)d ⩽ 0.288.Under only slightly stronger conditions (k(A)d ⩽ 0.25), it is shown in Section 7, using the Banach contraction theorem, that the iteration converges and the solution is locally unique.Section 8 contains a numerical example