  Stationary iterative methods with a symmetric splitting matrix are performed
as inner-iteration preconditioning for Krylov subspace methods. We give
conditions such that the inner-iteration preconditioning matrix is definite,
and show that conjugate gradient (CG) method preconditioned by the inner
iterations determines a solution of symmetric and positive semidefinite linear
systems, and the minimal residual (MINRES) method preconditioned by the inner
iterations determines a solution of symmetric linear systems including the
singular case. These results are applied to the CG and MINRES-type methods such
as the CGLS, LSMR, and CGNE methods preconditioned by inner iterations, and
thus justify using these methods for solving least squares and minimum-norm
solution problems whose coefficient matrices are not necessarily of full rank.
Thus, we complement the convergence theories of these methods presented in [K.
Morikuni and K. Hayami, SIAM J. Matrix Appl. Anal., 34 (2013), pp.1-22], [K.
Morikuni and K. Hayami, SIAM J. Matrix Appl. Anal., 36 (2015), pp. 225-250],
and give bounds for these methods.
