Augmentation of the Generalised n×n Eigenvalue Equation to a Generalised (n+1)×(n+1) Eigenvalue Equation

Abstract

Generalised n×n eigenvalue equation B|Φ _i 〉=λ _i S ^b|Φ _i 〉 (i=1,...,n) where B and S ^b are n×n Hermitian matrices while S ^b is in addition positive definite is considered. This equation is augmented to a generalised (n+1)(n+1) eigenvalue equation H|Ψ _k 〉=ε _k S|Ψ _k 〉 (k=1,...,n+1) where Hermitian matrices H and S represent matrices B and S ^b, respectively, augmented by one additional row and one additional column. It is shown how the eigenvalues ε _k and the eigenvectors |Ψ _k 〉 of the augmented eigenvalue equation can be expressed in terms of the eigenvalues λ _i and the eigenvectors |Φ _i 〉 of the original eigenvalue equation. Operation count to obtain by this method all augmented eigenvalues and eigenvectors is of the order O(n ²). Unless matrices involved are of some special kind such as sparse matrices or alike, this operation count is one order of magnitude smaller than operation count required by other presently known methods. In many practical cases operation count to obtain a single selected eigenvalue and/or eigenvector by this method is of the order O(n). In the case of the generalised eigenvalue equation, all other methods usually require again O(n ³) operations, even if only a single eigenvalue and/or eigenvector is required. Thus in many cases of interest operation count to obtain a selected eigenvalue and/or eigenvector by this method is two orders of magnitude smaller than operation count required by other methods.