eigenvalue problems

Show Summary Details

Quick Reference

Problems that arise frequently in engineering and science and fall into two main classes. The standard (matrix) eigenvalue problem is to determine real or complex numbers,

λ1, λ2,…λn (eigenvalues)

and corresponding nonzero vectors,

x1, x2,…, xn (eigenvectors)

that satisfy the equation

Ax = λx

where A is a given real or complex n×n matrix.

By analogy the continuous eigenvalue problem is to determine similar eigenvalues and corresponding nonzero functions (eigenfunctions) that satisfy the equation

Hf(x) = λf(x)

where H is a given operator on functions f. A simple example arising from a vibrating-string problem is

y″(x) = λy(x),

y(0) = 0, y(1) = 0

where values of the parameter λ (eigenvalues) are required that yield nontrivial eigenfunctions y(x) (i.e. y(x) ≢ 0). Finite-difference methods applied to such problems generally lead to matrix eigenvalue problems.

Subjects: Computing.

Reference entries

Users without a subscription are not able to see the full content. Please, subscribe or login to access all content.