Backward errors for the inverse eigenvalue problem (Q1294029)

For a class of inverse symmetric eigenvalue problems, where real numbers \(c_1,\dots, c_n\) are sought, such that \(A_0+ \sum^n_{k= 1} c_kA_k\), where \(A_k\) are symmetric \(n\times n\) matrices, have certain prescribed eigenvalues, a computable backward error is given, which bounds the norms of symmetric perturbation matrices \(\Delta A_k\) mainly by the deviation of the actual from the prescribed eigenvalues. This bound is further refined for the special case \(A_k= e_k e^T_k\), \(k= 1,\dots, n\). Detailled proofs are given for both bounds, and demonstrated with a numerical example.