Error Bounds for Computed Eigenvectors.

Keep the assignments to $\wtd\lambda$, $\lambda$, and $\delta$ as above, and let $x$ be the eigenvector of $A$ corresponding to $\lambda$. The error bound for the computed eigenvector $\wtd x$ and the true eigenvector $x$ are

$$\sin\theta_B(x,\wtd x)\le\frac 1{\delta} \cdot\frac {\Vert r\Vert _{B^{-1}}}{\Vert\wtd x\Vert _B}.$$ (97)

This, (5.27), and (5.28) yield

$$\sin\theta(x,\wtd x)\le\Vert B^{-1}\Vert _2\frac {\sqrt{2\kappa(B)}}{\delta}\Vert r\Vert _2.$$ (98)

The factor $\sqrt 2$ can be removed by a more elaborate argument, but we shall omit it here.