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We introduce a framework for defining a distance on the (non-Euclidean) space of Linear Dynamical Systems (LDSs). The proposed distance is induced by the action of the group of orthogonal matrices on the space of statespace realizations of LDSs. This distance can be efficiently computed for large-scale problems, hence it is suitable for applications in the analysis of dynamic visual scenes and other high dimensional time series. Based on this distance we devise a simple LDS averaging algorithm, which can be used for classification and clustering of time-series data. We test the validity as well as the performance of our group-action based distance on synthetic as well as real data and provide comparison with state-of-the-art methods.