Reformulating Generalized Eigenvalue Problems

The eigenvalue problem Ax = λ Bx is often difficult for users to analyze because it is frequently ill-conditioned. Occasionally, changes of variables can be performed on the given problem to ease this ill-conditioning. Suppose that B is singular, but A is nonsingular. Define the reciprocal \(\mu = \lambda^{-1}\). Then assuming A is definite, the roles of A and B are interchanged so that the reformulated problem Bx = μ Ax is solved. Those generalized eigenvalues \(\mu_j=0\) correspond to eigenvalues \(\lambda_j=\infty\). The remaining \(\lambda_j=\mu_j^{-1}\). The generalized eigenvectors for \(\lambda_j\) correspond to those for \(\mu_j\).

Now suppose that B is nonsingular. The user can solve the ordinary eigenvalue problem Cx = λ x, where \(C=B^{-1}A\). The matrix C is subject to perturbations due to ill-conditioning and rounding errors when computing \(B^{-1}A\). Computing the condition numbers of the eigenvalues for C may, however, be helpful for analyzing the accuracy of results for the generalized problem.

There is another method that users can consider to reduce the generalized problem to an alternate ordinary problem. This technique is based on first computing a matrix decomposition B=PQ, where both P and Q are matrices that are “simple” to invert. Then, the given generalized problem is equivalent to the ordinary eigenvalue problem Fy = λ y. The matrix \(F=P^{-1}AQ^{-1}\) and the unnormalized eigenvectors of the generalized problem are given by \(x=Q^{-1}y\).