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In this paper, we analyze the optimization landscape of gradient descent methods for static output feedback (SOF) control of discrete-time linear time-invariant systems with quadratic cost. The SOF setting can be quite common, for example, when there are unmodeled hidden states in the underlying process. We first establish several important properties of the SOF cost function, including coercivity, L-smoothness, and M-Lipschitz continuous Hessian. We then utilize these properties to show that the gradient descent is able to converge to a stationary point at a dimension-free rate. Furthermore, we prove that under some mild conditions, gradient descent converges linearly to a local minimum if the starting point is close to one. These results not only characterize the performance of gradient descent in optimizing the SOF problem, but also shed light on the efficiency of general policy gradient methods in reinforcement …