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 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 vibratingstring 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). Finitedifference methods applied to such problems generally lead to matrix eigenvalue problems.
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