Algebraic Eigenvalue Problem
H. M. Antia · Texts and readings in physical sciences · 2012
One of the important problems in linear algebra is the determination of eigenvalues and eigenvectors of a matrix. Eigenvalue problems in differential and integral equations can also be approximated by algebraic eigenvalue problems. In spite of the simplicity of its formulation, many algorithms are required to deal efficiently with a wide spectrum of problems, which are encountered in practice. The variety in problems arises because of the variety in type of matrix and the varying requirements from the solution. The classification of matrices into different types has been described in Section 3.1 and will not be repeated here. The varying requirements from the solution arise because we may need all eigenvalues and eigenvectors or only some of them. Alternately, we may be interested in only eigenvalues. In some problems, we may be interested in only the eigenvalue with the largest magnitude, or the one with the largest real part, while in other problems we may require all eigenvalues in a given region of the complex plane. In general, there is some correlation between the type of matrix and the amount of information required from the solution. For example, with large matrices which usually arise from discretisation of differential or integral equations, we are normally interested in only a few eigenvalues and possibly the eigenvectors. On the other hand, for small matrices we normally want all eigenvalues and some or all eigenvectors.