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.