# eigenvalue problems

**eigenvalue problems** 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*

**= λ**

*x*

*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*(

**) = λ**

*x**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.

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**eigenvalue problems**