Rectangular Multispectral Perturbation Theory

TL;DR

This paper advances rectangular multispectral perturbation theory by analyzing eigenvalues and eigenvectors, defining condition numbers and pseudospectra, and exploring computational and application implications.

Contribution

It extends perturbation theory to rectangular multiparameter eigenvalue problems, introducing new concepts and analysis tools for this complex setting.

Findings

Eigenvalues and eigenvectors are analyzed with new condition numbers.

Pseudospectra are defined for rectangular multiparameter problems.

Numerical examples link perturbation concepts and suggest optimal solutions relate to well-conditioned eigenvalues.

Abstract

We provide a first systematic treatment of so-called rectangular multispectral perturbation theory. With their paper from 2003, Hochstenbach and Plestenjak ["Backward Error, Condition Numbers, and Pseudospectra for the Multiparameter Eigenvalue Problem" in Linear Algebra and its Applications] extended perturbation theory from one-parameter eigenvalue problems to multiple spectral parameters. After two decades, we take it one step further and consider a different manifestation of the multiparameter eigenvalue problem that consists of one matrix equation with rectangular coefficient matrices. We perform a norm-wise backward error analysis, define condition numbers for both eigenvalues and eigenvectors, and introduce the pseudospectrum while also considering the computational implications of working with multiple spectral parameters. The rectangular shape hampers a direct application of the existing definitions and properties. For example, the left null space at a given eigenvalue is non-trivial and the dimensions of the left and right eigenvectors are different. Through numerical examples, we illustrate and link the different concepts from the perturbation theory. A system identification application seem to suggest that, in optimization-driven problems for which multiparameter reformulations exist, the globally optimal solutions tend to coincide with the best-conditioned eigenvalues.