Prove in newton's method

Question

I have to prove, that the direction in Newton's method is a descent direction if the Hessian is positive defnite.

My idea:

$ direction = -H(x)^{-1}*\nabla f(x)$

Put how can I prove that this is a descent direction?

Answer 1

By definition, a vector $p$ is a descent direction if $$ \nabla f(x) \cdot p <0.$$ Now, since $\nabla^2f(x)$ is positive definite, $\nabla^2f(x)^{-1}$ is also positive definite and therefore $$ \forall y, \; \; \; \nabla^2f(x)^{-1}y\cdot y>0.$$ We can then deduce that $$\nabla f(x) \cdot (-\nabla^2f(x) ^{-1} \nabla f(x)) = - (\nabla^2f(x)^{-1} \nabla f(x)) \cdot \nabla f(x)<0$$ which means that this is indeed a descent direction.