Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 41C / c / Solution

The gradient descent iteration isFor the error $e^{(k)}=x^{(k)}-A^{-1}b$,Part (a) shows that convergence for every initial vector is equivalent to
$|1-\alpha {\lambda }_i|<1$ for every eigenvalue of $A$. Since all eigenvalues are positive,