  We propose a conjugate gradient type optimization technique for the
computation of the Karcher mean on the set of complex linear subspaces of fixed
dimension, modeled by the so-called Grassmannian. The identification of the
Grassmannian with Hermitian projection matrices allows an accessible
introduction of the geometric concepts required for an intrinsic conjugate
gradient method. In particular, proper definitions of geodesics, parallel
transport, and the Riemannian gradient of the Karcher mean function are
presented. We provide an efficient step-size selection for the special case of
one dimensional complex subspaces and illustrate how the method can be employed
for blind identification via numerical experiments.
