Minimization principles and computation for the generalized linear response eigenvalue problem

The minimization principle and Cauchy-like interlacing inequalities for the generalized linear response eigenvalue problem are presented. Based on these theoretical results, the best approximations through structure-preserving subspace projection and a locally optimal block conjugate gradient-like algorithm for simultaneously computing the first few smallest eigenvalues with the positive sign are proposed. Numerical results are presented to illustrate essential convergence behaviors of the proposed algorithm.