We analyse an iterative algorithm to minimize quadratic functions whose Hessian matrix $H$ is the expectation of a random symmetric $d\times d$ matrix. The algorithm is a variant of the stochastic variance reduced gradient (SVRG). In several applications, including least-squares regressions, ridge regressions, linear discriminant analysis and regularized linear discriminant analysis, the running time of each iteration is proportional to $d$. Under smoothness and convexity conditions, the algorithm has linear convergence. When applied to quadratic functions, our analysis improves the state-of-the-art performance of SVRG up to a logarithmic factor. Furthermore, for well-conditioned quadratic problems, our analysis improves the state-of-the-art running times of accelerated SVRG, and is better than the known matching lower bound, by a logarithmic factor. Our theoretical results are backed with numerical experiments.