Gradients of Quotients and Eigenvalue Problems

TL;DR

This paper explores the connection between quotients, eigenvalue problems, and optimization, introducing new classes of nonlinear eigenvalue problems and tools for their analysis, including generalized and $p$-norm-based formulations.

Contribution

It develops a framework for analyzing nonlinear eigenvalue problems derived from quotients, including generalized, $p$-norm, and homogeneous cases, with new notions and tools for their classification.

Findings

Generalized folded spectrum eigenvalue problem for Hermitian matrices.

Introduction of $p$-Laplacian eigenvalue problem for matrices.

Development of tools to identify gradient eigenvalue problems.

Abstract

Intertwining analysis, algebra, numerical analysis and optimization, computing conjugate co-gradients of real-valued quotients gives rise to eigenvalue problems. In the linear Hermitian case, by inspecting optimal quotients in terms of taking the conjugate co-gradient for their critical points, a generalized folded spectrum eigenvalue problem arises. Replacing the Euclidean norm in optimal quotients with the $p$-norm, a matrix version of the so-called $p$-Laplacian eigenvalue problem arises. Such nonlinear eigenvalue problems seem to be naturally classified as being a special case of homogeneous problems. Being a quite general class, tools are developed for recovering whether a given homogeneous eigenvalue problem is a gradient eigenvalue problem. It turns out to be a delicate issue to come up with a valid quotient. A notion of nonlinear Hermitian eigenvalue problem is suggested. Cauchy-Schwarz quotients are introduced.