An iteration process for obtaining small eigenvalues of a matrix and the corresponding eigenvectors (Q2544888)

In this paper the author describes an algorithm for obtaining matrix eigenvectors corresponding to eigenvalues which are near zero and which belong to linear elementary divisors. The method is too complicated to present in detail. Essentially it uses the triangularization of a given \(n\times n\) matrix by an orthogonal matrix to obtain the number, \(n - r\), of ``small'' eigenvalues of the given matrix. Any such eigenvalue and its eigenvector may be represented as infinite series whose terms can be determined recursively. As part of the recursion it is necessary either to solve the complete eigenvalue problems of a certain \((n - r)\times (n - r)\) matrix and its transpose, or to perform auxiliary triangular decompositions an them. Thus, the method is best suited for problems where \(n - r\) is small, as, for example, in the refinement of a previously calculated eigenvalue when, most likely, \(n - r = 1\).