= Steepest-descent step size
{title2=$\alpha=\frac{r^Tr}{r^TAr}$}
For a real <symmetric positive-definite matrix> $A$ and nonzero <linear-system residual> $r=b-Ax$, minimize $f(x+\alpha r)$ for $f(x)=x^TAx/2-b^Tx$. Differentiating with respect to $\alpha$ gives $-r^Tr+\alpha r^TAr=0$, 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.
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