Neuromorphic architectures achieve low-power operation by using many simple spiking neurons in lieu of traditional hardware. Here, we develop methods for precise linear computations in spiking neural networks and use these methods to map the evolution of a linear dynamical system (LDS) onto an existing neuromorphic chip: IBM's TrueNorth. We analytically characterize, and numerically validate, the discrepancy between the spiking LDS state sequence and that of its non-spiking counterpart. These analytical results shed light on the multiway tradeoff between time, space, energy, and accuracy in neuromorphic computation. To demonstrate the utility of our work, we implemented a neuromorphic Kalman filter (KF) and used it for offline decoding of human vocal pitch from neural data. The neuromorphic KF could be used for low-power filtering in domains beyond neuroscience, such as navigation or robotics.