## 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,

λ_{1}, λ_{2},…λ* _{n}* (

**eigenvalues**)

and corresponding nonzero vectors,

*x*_{1}, *x*_{2},…, *x** _{n}* (

**eigenvectors**)

that satisfy the equation

*A**x* = λ*x*

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*) = λ*f*(*x*)

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

*y*″(*x*) = λ*y*(*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.