On an integrable discretization of the Rayleigh quotient gradient system and the power method with a shift

An integrable discretization of the Rayleigh quotient gradient system is established. The solution of the discrete gradient system is described explicitly and converges exponentially to the same equilibrium point as that of the continuous gradient system for arbitrary large difference step size. It is shown that the discrete gradient system is essentially equivalent to the power method with a shift of origin for calculating the largest eigenvalue. The power method is then proved to be a discrete gradient method.