The conjugate gradient algorithm applied to quaternion - valued matrices

The well known conjugate gradient algorithm (cg-algorithm), introduced by Hestenes & Stiefel, [1952] intended for real, symmetric, positive definite matrices works as well for complex matrices and has the same typical convergence behavior. It will also work, not generally, but in many cases for hermitean, but not necessarily positive definite matrices. We shall show, that the same behavior is still valid if we apply the cg-algorithm to matrices with quaternion entries. We particularly investigate the early stop of the cg-algorithm in this case and we develop error estimates. We have to present some basic facts about quaternions and about matrices with quaternion entries, in particular, about eigenvalues of such matrices. We also present some numerical examples of quaternion systems solved by the cg-algorithm.