  It is well-known that the convergence of Krylov subspace methods to solve
linear system depends on the spectrum of the coefficient matrix, moreover, it
is widely accepted that for both symmetric and unsymmetric systems Krylov
subspace methods will converge fast if the spectrum of the coefficient matrix
is clustered. In this paper we investigate the spectrum of the system
preconditioned by the deflation, coarse grid correction and adapted deflation
preconditioners. Our analysis shows that the spectrum of the preconditioned
system is highly impacted by the angle between the coarse grid space for the
construction of the three preconditioners and the subspace spanned by the
eigenvectors associated with the small eigenvalues of the coefficient matrix.
Furthermore, we prove that with a certain restriction the accuracy of the
inverse of projection matrix also impacts the spectrum of the preconditioned
system. Numerical experiments emphasized the theoretical analysis.
