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

T(λ)x=0(60.1)

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

It generalizes the linear eigenvalue problem Ax = λx, A ∈ ℂn×n, where T(λ) = λIA, and the generalized linear eigenvalue problem where T(λ) = λBA, 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 ...

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