Rayleigh Quotient Based Optimization Methods For Eigenvalue Problems

Four classes of eigenvalue problems that admit similar min-max principles and the Cauchy interlacing inequalities as the symmetric eigenvalue problem famously does are investigated. These min-max principles pave ways for efficient numerical solutions for extreme eigenpairs by optimizing the so-called Rayleigh quotient functions. In fact, scientists and engineers have already been doing that for computing the eigenvalues and eigenvectors of Hermitian matrix pencils A − λB with B positive definite, the first class of our eigenvalue problems. But little attention has gone to the other three classes: positive semidefinite pencils, linear response eigenvalue problems, and hyperbolic eigenvalue problems, in part because most min-max principles for the latter were discovered only very recently and some more are being discovered. It is expected that they will drive the effort to design better optimization based numerical methods for years to come.