Minimization Principles for the Linear Response Eigenvalue Problem I: Theory

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.