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.
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