The vertical and horizontal forces and associated stiffnesses on a permanent magnet (PM) above a high-temperature superconductor (HTS) were measured during vertical and horizontal traverses in zero-field cooling (ZFC) and in field cooling (FC). In ZFC, the vertical stiffness was greater in the first descent than in the first ascent and second descent, and the stiffness in the second descent was between those of the first descent and the first ascent. At the FC position, the vertical stiffness was two times greater than the lateral stiffness at each height, to within 1% of the vertical stiffness value. The cross stiffness of vertical force with respect to lateral position was positive for FC, but negative for ZFC. Free-spin-down experiments of a PM levitated above a HTS were also performed. These results showed that the coefficient of friction is double valued at frequencies just below the rotor resonance, a result attributed to cross stiffness in the PM/HTS interaction. A frozen-image model was used to calculate the vertical and horizontal forces and stiffnesses, and reasonable agreement with the data occurred for vertical or horizontal movements of the PM less than several mm from the FC position. (C) 1999 American Institute of Physics. [S0021-8979(99)03223-5].