Well, Mr. Jacobi, here it is: all the generalized inversion of two generations of invertors who, knowingly or unknowingly, subscribed to (and extended) your dictum. Please forgive us if we have over-inverted, or if we have not always inverted in the natural and sensible way. (Some of us have inverted with labor and pain by using hints from a dean or a tenure and promotion committee that “you better invert more, or else you would be inverted.”)

Annotated Bibliography on Generalized Inverses and Applications

LetA andB ben×m matrices. A gradient method for the minimization of the functionalF(x)=?Ax?(?Ax, Bx?/?Bx, Bx?)Bx?2 is developed. The minima ofF are the eigenvectors of the eigenproblemAx=?Bx. The concept of a non-defective eigenvalue for this generalized eigenvalue problem is developed. It is then shown that geometric convergence is attained for non-defective eigenvalues. A convergence rate analysis is given where it is shown that the rapidity of convergence of the gradient method to an eigenvalue ? depends on the degree of non-defectiveness of ? and the singular values ofA??B.

A gradient method for the matrix eigenvalue problemAx=λBx

A uniform approach to gradient methods for linear operator equations

A general convergence principle in nonlinear functional analysis

A gradient method is presented for solving the eigentuple-eigenvector problem, $$Ax = \sum\limits_{i = 1}^P {\lambda _i B_i x} $$ which is a generalization ofAx=?Bx that occurs in generalized Sturm-Liouville theory. Under certain conditions on matricesA, B 1,...,B P , the method is proved to converge locally. Numerical computations are given in a companion paper.

G. H. Rodrigue

Papers

Solution of eigenvalue problems in Hilbert spaces by a gradient method