Exact line search for a positive-definite quadratic

For $f(x)=x^{T}Ax/2-b^{T}x$ with $A$ positive definite and residual $r=b-Ax$, exact line search along $r$ usesThe objective error contracts by

For a real symmetric positive-definite matrix $A$ and nonzero linear-system residual $r=b-Ax$, minimize $f(x+\alpha r)$ for $f(x)=x^{T}Ax/2-b^{T}x$. Differentiating with respect to $\alpha$ gives $-r^{T}r+\alpha r^{T}Ar=0$, yielding the displayed positive step. Positive definiteness makes it the unique minimum along the line. Applying the same quotient to a nonsymmetric or indefinite matrix is a formal iteration, but this minimization proof and its convergence consequences no longer apply.