5 - Eigenvalue Problems

Abstract

This chapter focuses on eigenvalue problems. The determination of frequencies in freely oscillating mechanical or electrical systems, or the determination of critical frequencies for rotating shafts and similar technical questions, lead to eigenvalue problems. Eigenvalue problems belong to the field of nonlinear algebra. The eigenvalue problem consists in determining the unknown parameter λ in such a way that the homogeneous equations have a nontrivial solution. An eigensolution is determined only up to a proportionality factor p. It is possible that corresponding to an eigenvalue λ, there exist two eigensolutions that are not proportional to each other. In that case, the eigenvalue is called degenerate. Premature stoppage prevents a clear and exhaustive solution of the eigenvalue problem. A symmetric eigenvalue problem can have only real eigenvalues. To calculate the eigenvalues of the problem, the poles of the resolvent are determined by means of the Bernoulli method.