In this paper, we propose a new local spatio-temporal descriptor for videos and we propose a new approach for action recognition in videos based on the introduced descriptor. The new descriptor is called the Video Covariance Matrix Logarithm (VCML). The VCML descriptor is based on a covariance matrix representation, and it models relationships between different low-level features, such as intensity and gradient. We apply the VCML descriptor to encode appearance information of local spatio-temporal video volumes, which are extracted by the Dense Trajectories. Then, we present an extensive evaluation of the proposed VCML descriptor with the Fisher vector encoding and the Support Vector Machines on four challenging action recognition datasets. We show that the VCML descriptor achieves better results than the state-of-the-art appearance descriptors. Moreover, we present that the VCML descriptor carries complementary information to the HOG descriptor and their fusion gives a significant improvement in action recognition accuracy. Finally, we show that the VCML descriptor improves action recognition accuracy in comparison to the state-of-the-art Dense Trajectories, and that the proposed approach achieves superior performance to the state-of-the-art methods.