Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 40C / e / Solution

Let the computed eigenpair residual beDefine the rank-one matrixThenso $(\tilde{\lambda },\tilde{v})$ is an exact eigenpair of the perturbed matrix $A+E$. Moreover,which is the backward error of an approximate eigenpair.

Follow the simple eigenvalue branch from $\lambda (0)=\lambda$ for $A(t)=A+tE$. The first-order perturbation of a simple eigenvalue and part (b) giveSince the residual is at the machine precision scale, this proves