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Show that the symmetric rank-one update (2.18) and the BFGS update (2.19) are scale-invariant if the initial Hessian approximations B0 are chosen appropriately. That is, using the notation of the previous exercise, show that if these methods are applied to f (x) starting from x0 Sz0 + s with initial Hessian B0, and to ˜f (z) starting from z0 with initial Hessian ST B0S, then all iterates are related by x_{k} =S_{zk} + s. (Assume for simplicity that the methods take unit step lengths.)
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