Calculation of eigenvalues

Calculating the eigenvalues and eigenvectors of matrices is one of the most important problems in linear numerical analysis. Techniques requiring knowledge of the spectrum of matrices are used in fields as varied as quantum mechanics, structure analysis, graph theory, economic models and the ranking of pages on the World Wide Web by search engines.

For example, in structural mechanics, the problems of "resonances" or "vibrations" of mechanical structures, described by spectral analysis, boil down to calculations of eigenvalues and eigenvectors.

Non-symmetrical eigenvalue problems arise in the stability analysis of dynamical systems. In a completely different field, quantum chemistry gives rise to symmetrical eigenvalue problems that can be gigantic, both in size and in the number of eigenvalues and eigenvectors to be extracted. Singular-value decomposition, which is a kind of generalization of classical spectral decomposition, is also of prime importance in statistics and "new economy" problems (pattern recognition, data mining, signal processing, data mining, etc.).

Eigenvalue problems are very rich, both in terms of their variety and in terms of the type of matrices to be dealt with and the calculation methods and algorithms to be used: matrices can be symmetric or non-symmetric, hollow or solid, and problems can be classical or generalized or even quadratic. Some applications require the calculation of a very small number of eigenvalues, while others require a large number of eigenvalues or even the entire spectrum.

In this article, we'll try to give you an overview of the tools available for solving these different cases.