Adversarial domain adaptation has made remarkable advances in learning transferable representations for knowledge transfer across domains. While adversarial learning strengthens the feature transferability which the community focuses on, its impact on the feature discriminability has not been fully explored. In this paper, a series of experiments based on spectral analysis of the feature representations have been conducted, revealing an unexpected deterioration of the discriminability while learning transferable features adversarially. Our key finding is that the eigenvectors with the largest singular values will dominate the feature transferability. As a consequence, the transferability is enhanced at the expense of over penalization of other eigenvectors that embody rich structures crucial for discriminability. Towards this problem, we present Batch Spectral Penalization (BSP), a general approach to penalizing the largest singular values so that other eigenvectors can be relatively strengthened to boost the feature discriminability. Experiments show that the approach significantly improves upon representative adversarial domain adaptation methods to yield state of the art results.