Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof / 6 / a / iv

Write $D=\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_m)$ and let $|\Lambda |={\max }_i|{\lambda }_i|$. The stated Euclidean norm givesTaking nonnegative square roots proves $\Vert f_n\Vert \le |\Lambda {|}^n\Vert f_0\Vert$. At $n=0$ this is equality. This is a contraction bound in the transformed coordinates; an arbitrary nonorthogonal $S$ does not give the same unit-constant bound for the original error norm.