On investigating GMRES convergence using unitary matrices

For a given matrix A and right-hand side b, this paper investigates unitary matrices generating, with some right-hand sides c, the same GMRES residual norms as the pair (A,b). We give characterizations of this class of unitary matrices and point out the relationship with Krylov subspaces and Krylov residual subspaces for the pair (A,b). We investigate the eigenvalues of these unitary matrices in relation to the convergence behavior of GMRES for the pair (A,b) and describe the indispensable role of the eigenvector information. We conclude with a formula for the GMRES residual norms generated by a normal matrix B in terms of its eigenvalues and components of the right-hand side c in the eigenvector basis.