This paper is concerned with proving theoretical results related to the convergence of the conjugate gradient (CG) method for solving positive definite symmetric linear systems. Considering the inverse of the projection of the inverse of the matrix, new relations for ratios of the A-norm of the error and the norm of the residual are provided, starting from some earlier results of Sadok (Numer. Algorithms 2005; 40:201–216). The proofs of our results rely on the well-known correspondence between the CG method and the Lanczos algorithm. Copyright © 2008 John Wiley & Sons, Ltd.