We propose an approach to use the state covariance of linear systems to track time-varying covariance matrices of non-stationary time series. Following concepts from Riemmanian geometry, we investigate three types of covariance paths obtained by using different quadratic regularizations of system matrices. The first quadratic form induces the geodesics based on the Bures-Wasserstein metric from optimal mass transport theory and quantum mechanics. The second type of quadratic form leads to the geodesics based on the Fisher-Rao metric from information geometry. In the process, we introduce a fluid-mechanics interpretation of the Fisher-Rao metric for multivariate Gaussian distributions. A main contribution of this work is the introduction of the third type of covariance paths which are steered by linear system matrices with rotating eigenspace. We provide theoretical results on the existence and uniqueness of this type of covariance paths. The three types of covariance paths are compared using two examples with synthetic data and real data based on functional magnetic resonance imaging, respectively.