Adaptive estimation and identification algorithms involving unknown symmetric and positive definite (SPD) matrix-valued parameters are ubiquitous in engineering applications. The problem of estimating the noise covariance matrices in estimation algorithms is considered first. An adaptive Kalman filter to estimate the noise covariance matrix of the noises entering a linear time invariant system is introduced first. The convergence of the estimates as well as the states is guaranteed with mild assumptions on the system. Conditions of estimability of the noise covariance matrix are discussed. The generalization of the adaptive Kalman fitler to the linear time varying case is introduced next. To maintain positive definiteness of the noise covariance estimates a differential geometric approach is adopted. The geometry of the manifold of SPD matrices is used to develop a Riemannian optimization based adaptive Kalman filter that ensure positive definiteness of the estimate. The convergence of the Riemannian optimization-based estimate and the adaptive Kalman filter is established under mild conditions of uniform observability and uniform controllability of the system. An adaptive control problem with an unknown SPD matrix is considered next. A novel projection scheme is introduced that ensures that the estimates of the unknown SPD matrix are SPD. Adaptive update laws for identifying the SPD matrix are also presented. The adaptive control laws are shown to globally stabilize systems in problems such as the adaptive angular velocity tracking, adaptive attitude control, and the adaptive trajectory tracking of robotic manipulators with parameter uncertainties within the generalized mass matrix. In general, such a method can be applied to estimation of symmetric matrices with eigenvalue constraints.Aerospace Engineerin
