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For a fixed shift $s$ that is not an eigenvalue, inverse iteration is the power method applied to $(A-sI{)}^{-1}$. Before normalization, the $k$th vector is proportional toIf $s<{\lambda }_1$, factor out the first coefficient:Every ratio in parentheses lies strictly between zero and one. The bracket therefore tends to $w_1$, and normalization in the Euclidean norm givesThis proves the stated convergence to $w_1$ or $-w_1$ and exhibits the convergence of fixed-shift inverse iteration rate.

Now suppose ${\lambda }_m<s<{\lambda }_{m+1}$. The closest eigenvalue to $s$ is one of ${\lambda }_m$ and ${\lambda }_{m+1}$.

Finally, if ${\lambda }_n<s$, the closest eigenvalue is ${\lambda }_n$ and its inverse eigenvalue is negative. Thus the direction approaches the line spanned by $w_n$, while normalized vectors alternate between the two signs. These cases are the sign behavior of fixed-shift inverse iteration.

Fix a shift $s$ that is not an eigenvalue of $A$. Inverse iteration computesand may use $r({\mathbf{x}}_k)$ as the eigenvalue estimate.

For a real symmetric matrix, suppose that one eigenvalue ${\lambda }_{\ast }$ is uniquely closest to $s$ and that ${\mathbf{x}}_0$ has a nonzero component in its eigenspace. Since the iteration is the power method applied to $(A-sI{)}^{-1}$, the normalized vectors converge, up to sign, to an eigenvector for ${\lambda }_{\ast }$. If ${\lambda }_{\mathrm{next}}$ is the second-closest eigenvalue to the shift, the asymptotic direction-error ratio isThus a shift close to the desired eigenvalue gives rapid convergence. The Rayleigh quotients converge to ${\lambda }_{\ast }$ and, for a simple eigenvalue, their errors are of second order in the vector error. This is convergence of fixed-shift inverse iteration.