A note on spectrum analysis of augmentation block preconditioned generalized saddle point matrices

In this note we consider a set of augmentation block preconditioners for solving generalized saddle point systems whose coefficient matrices have singular (1,1) blocks. Results concerning the eigenvalue distribution and forms of the eigenvectors of the augmentation block preconditioned generalized saddle point matrix and its minimal polynomial are given, and an optimal augmentation block preconditioner of the set is derived. These results extend previous ones in the literature. Numerical experiments that show the very effective performance of the optimal augmentation block preconditioner are reported.