Eigenvalue Problems

Abstract

Eigenvalue problems are very common in physics. In many cases they involve solution of a homogeneous system of linear equations with a Hermitian (or symmetric, if real) matrix. The direct solution of the eigenvalue problem is only possible for matrices of very small dimension. For medium-sized problems the Jacobi method or reduction to tridiagonal form by a series of Householder reflections are appropriate. Special algorithms are available for matrices of very large dimension to calculate a small number of eigenvalues and eigenvectors. The famous Lanczos is discussed. A computer experiment demonstrates the application to disorder in a two-dimensional tight-binding model.