The timely precoding of frequency division duplex (FDD) massive multiple-input multiple-output (MIMO) systems is a substantial challenge in practice, especially in mobile environments. In order to improve the precoding performance and reduce the precoding complexity, we propose a partial reciprocity-based precoding matrix prediction scheme and further reduce its complexity by exploiting the channel gram matrix. We prove that the precoders can be predicted through a closed-form eigenvector interpolation which was based on the periodic eigenvector samples. Numerical results validate the performance improvements of our schemes over the conventional schemes from 30 km/h to 500 km/h of moving speed.