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PutThe Rayleigh quotient shift is exactlyExpanding the denominator givesThe question prints this as $s_k={\lambda }_1-K{ϵ}^2+O({ϵ}^4)$, so its sign convention requiresunless every $a_i$ vanishes. Equivalently, $-K$ is the positive quadratic Rayleigh-quotient error.

The shifted inverse solve has the spectral decompositionOnly relative coefficients matter after normalization, so factor out $({\lambda }_1-s_k{)}^{-1}$:From the shift expansion,whereas, for $i\ge 2$,ConsequentlyAfter the final normalization,Thus an $O(ϵ)$ eigenvector-direction error becomes $O({ϵ}^3)$ in one step, proving the Local cubic convergence of Rayleigh quotient iteration for this symmetric matrix.