We propose an approach to use the state covariance of autonomous linear systems to track time-varying covariance matrices of nonstationary time series. Following concepts from the Riemannian 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 Hellinger-Bures metric related to optimal mass transport (OMT) 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 weighted-OMT 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 system matrices with rotating eigenspaces. The three types of covariance paths are compared using two examples with synthetic data and real data from resting-state functional magnetic resonance imaging, respectively.