  We report on the experimental study of an exceptional point (EP) in a
dissipative microwave billiard with induced time-reversal invariance (T)
violation. The associated two-state Hamiltonian is non-Hermitian and
non-symmetric. It is determined experimentally on a narrow grid in a parameter
plane around the EP. At the EP the size of T violation is given by the relative
phase of the eigenvector components. The eigenvectors are adiabatically
transported around the EP, whereupon they gather geometric phases and in
addition geometric amplitudes different from unity.
