On the theory of deflation and singular symmetric positive semi-definite matrices

In this report we give new insights into the properties of invertible and singular deflated and preconditioned linear systems where the coefficient matrices are also symmetric and positive (semi-) definite. First we prove that the invertible de ated matrix has always a more favorable effeective condition number compared to the original matrix. So, in theory, the solution of the deflated linear system converges faster in iterative methods than the original one. Thereafter, some results are presented considering the singular systems originally from the Poisson equation with Neumann boundary conditions. In practice these linear systems are forced to be invertible leading to a worse (eective) condition number. We show that applying the deflation technique remedies this problem of a worse condition number. Moreover, we derive some useful equalities between the de ated variants of the singular and invertible matrices. Then we prove that the de ated singular matrix has always a more favorable effective condition number compared by the original matrix