Tove Odland: On the relationship between quasi-Newton methods and the conjugate gradient method when minimizing a quadratic function

It is well known that a Quasi-Newton method using any well-defined update from the Broyden class of updates and the conjugate gradient method produce the same iterates on a quadratic objective function with positive-definite Hessian. In this case both methods produce conjugate directions with respect to the Hessian. This equivalence does not hold for any quasi-Newton method. We discuss more precisely what the updates in a Quasi-Newton method need satisfy to gives rise to this behavior.