Gradients of Quotients and 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 -norm, a matrix version of the so-called -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.
Cite
@article{arxiv.2211.06051,
title = {Gradients of Quotients and Eigenvalue Problems},
author = {Marko Huhtanen and Olavi Nevanlinna},
journal= {arXiv preprint arXiv:2211.06051},
year = {2022}
}