Minimization Principles for the Linear Response Eigenvalue Problem I: Theory

Abstract

We present two theoretical results for the linear response eigenvalue problem. The first result is a minimization principle for the sum of the smallest eigenvalues with the positive sign. The second result is Cauchy-like interlacing inequalities. Although the linear response eigenvalue problem is a nonsymmetric eigenvalue problem, these results mirror the well-known trace minimization principle and Cauchy's interlacing inequalities for the symmetric eigenvalue problem.

Rayleigh--Ritz Approximation For the Linear Response Eigenvalue Problem

“…The eigenvalue problem (1.1) is referred to as the Linear Response Eigenvalue Problem (LREP) in the literature of computational quantum chemistry and physics, and several minimization principles were recently established and, as a result, CG type optimization algorithms were proposed to solve (1.1) [2,3,15]. Using the symmetric orthogonal matrix which was still referred to as the linear response eigenvalue problem (LREP) [2,16,23] and will be in this paper, too. The condition (1.2) implies that both K and M are symmetric and positive definite [2].…”

“…Using the symmetric orthogonal matrix which was still referred to as the linear response eigenvalue problem (LREP) [2,16,23] and will be in this paper, too. The condition (1.2) implies that both K and M are symmetric and positive definite [2]. Denote by In [2], a subspace version of this for characterizing the k smallest positive eigenvalues…”

“…The condition (1.2) implies that both K and M are symmetric and positive definite [2]. Denote by In [2], a subspace version of this for characterizing the k smallest positive eigenvalues…”

“…An important notion for LREP [2] is the so-called pair of deflating subspaces {U, V} by which we mean that both U and V are subspaces of R n and satisfy KU ⊆ V and M V ⊆ U.…”

“…In computational quantum chemistry and physics, the so-called random phase approximation (RPA) describes the excitation states (energies) of physical systems in the study of collective motion of many-particle systems [2,20,21], which has applications in silicon nanoparticles and nanoscale materials and analysis of interstellar clouds [2,3]. One important question in RPA is to compute a few eigenpairs associated with the smallest positive eigenvalues of the following eigenvalue problem:…”