A new convergence rate of the steepest descent regarding the Euclidean norm

We analyze the convergence rate of the steepest descent method, regarding the Euclidean norm with exact line search, applied to solve the problem of unconstrained minimization of strongly convex quadratic functions. We present an improved bound for the convergence rate presented in the literature, which provides a highly accurate estimate for the exact Q-linear convergence rate of the method. Moreover, we study the behavior of the exact rate, establishing some expected and natural properties, as the dependence only on the condition number of the Hessian and its monotonicity. We prove that the general analysis can be carried out by treating the two-dimensional case.