The authors study the convergence analysis of nonsymmetric linear systems in the context of the generalized minimal residual (GMRES) iteration and compare it with the abstract nonsymmetric balancing preconditioner. They show that many results for symmetric positive definite matrices carry over to arbitrary non-symmetric matrices. They establish that the spectra of the preconditioned systems are similar. They show that under certain conditions, the \(2\)-norm of residuals produced by GMRES combined with deflation is never larger than the \(2\)-norm of residuals produced by GMRES combined with the abstract balancing preconditioner. Numerical examples that confirm the theoretical results are also provided.