Show for a symmetric, positiv semi definite matrix $A$, a vector $b\in Ran(A)$ and initial vector $x_0$:
(1) All directions $d_0,d_1,...,d_m$ of the conjugate gradient method are in the range of $A$.
(2) For all $j\in\{0,...,m\}$ with $Ad_j\ne 0$ we have $d_j^TAd_j>0$.
(1) was no problem for me using the update equations for $d_j$. (2) says the CG-method is usable in the positive semidefinite case. But I am having trouble getting started with this one.
Edit: Using (1) we have that $d_j=Ay$ for some $y$. Then $d_j^TAd_j=(Ay)^TAd_j=yA^TAd_j$. I don't know if that helps me though.