On computational methods for the generalized eigenvalue problem

A considerable amount of material has been published on procedures for the solution of the standard eigenproblem Ax = [lambda]x. In the last few years increasing attention has been focused on the more general problem Ax = [lambda]Bx. This study of the generalized eigenproblem Ax = [lambda]Bx can be divided into three main parts. The first part is concerned with the solution of this problem when A and B are certain types of nonsingular matrices such as elementary, diagonal, or triangular. A survey of the numerical methods used to solve the eigenproblem Ax = [lambda]Bx when A and B are symmetric and B is positive definite is given in the second part. The third part uses the pseudoinverse of a matrix to solve Ax = [lambda]Bx when A and B are singular.