Handbook of Linear Algebra, 2nd Edition

Chapter 60

Nonlinefr Eigenvalue Problems

Heinrich Voss

Hamburg University of Technology

This chapter considers the nonlinear eigenvalue problem to find a parameter λ such that the linear system

$\begin{array}{cc}T\left(\lambda \right)x\text{=0}& \left(60.1\right)\end{array}$

has a nontrivial solution x, where T(·) : D → ℂn×n is a family of matrices depending on a complex parameter λ ∈ D.

It generalizes the linear eigenvalue problem Ax = λx, A ∈ ℂn×n, where T(λ) = λI − A, and the generalized linear eigenvalue problem where T(λ) = λB − A, A, B ∈ ℂn×n.

Nonlinear eigenvalue problems T(λ)x = 0 arise in a variety of applications in science and engineering, such as the dynamic analysis of structures, vibrations of fluid-solid structures, the electronic behavior of quantum dots, and delay eigenvalue problems, to name ...