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For a nonzero real vector $x$, the Rayleigh quotient of the matrix $A$ isThe Rayleigh quotient iteration starts from a unit vector $v_0$. Given $v_k$, compute the shiftsolve the linear systemand normalize in the Euclidean norm:The next eigenvalue estimate is ${\mu }_{k+1}=R_A(v_{k+1})$. The process stops when the eigenpair residual is sufficiently small.

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