Unit sphere-constrained quadratic optimization has been studied extensively over the past decades. While state-of-art algorithms for solving this problem often rely on relaxation or approximation techniques, there has been little research into scalable first-order methods that tackle the problem in its original form. These first-order methods are often more well-suited for the big data setting. In this paper, we provide a novel analysis of the simple projected gradient descent method for minimizing a quadratic over a sphere. When the gradient step size is sufficiently small, we show that convergence is locally linear and provide a closed-form expression for the rate. Moreover, a careful selection of the step size can stimulate convergence to the global solution while preventing convergence to local minima.