In a large number of applications one needs to solve a linear system whose associated matrix can be introduced as a 2x2 block matrix with a zero (2,2) block. Such linear systems arise in constrained optimization, least squares problems, the discrete Stokes equation, and other areas of applications.

In this talk we discuss solution techniques for solving such systems, with focus on situations where the (1,1) block is possibly singular. In such cases straightforward Schur complement techniques cannot be applied. Two

different methods are described: a deflation approach, and an augmented Lagrangian technique. The first approach is based on identifying the nullity of the (1,1) block and eliminating a small number of unknowns so as to generate a 'reduced' indefinite system with a nonsingular (1,1) block.

In contrast, the augmented Lagrangian technique does not reduce the size of the linear system; rather, it modifies the (1,1) block so as to eliminate a possible singularity. As some of our analytical observations

and numerical examples illustrate, the numerical properties of the resulting linear system are considerably different than those of the original one.

This is joint work with Gene Golub at Stanford University.