 Solution

The exact minimization gives $r_{j+1}=r_j-{\alpha }_{j}A{p}_j$, with ${\alpha }_j=p_j^T{r}_j/(p_j^{T}A{p}_j)$. For steepest descent method, the next search direction is $p_{j+1}=r_{j+1}$. HenceThus consecutive nonzero search directions are orthogonal. This calculation even covers an arbitrary initial direction $p_1$, as in the generic algorithm: it only needs the exact preceding line minimization and the choice $p_{j+1}=r_{j+1}$. If the new linear-system residual is zero, the linear system is already solved. No pairwise orthogonality of all nonconsecutive directions is asserted.