Inverse iteration

Remarks

The matrix $(A - \mu I)^{-1}$ has the same eigenvectors as $A$, with eigenvalues $1/(\lambda_i - \mu)$. The hypothesis makes $1/(\lambda_j - \mu)$ the eigenvalue of largest modulus, so inverse iteration is power iteration applied to $(A - \mu I)^{-1}$. Expanding $(A - \mu I)^{-k} x_0 = \sum_i c_i (\lambda_i - \mu)^{-k} v_i$ and dividing by $(\lambda_j - \mu)^{-k}$ shows that the other components shrink like $\rho^k$.

Example: for $A = \mathrm{diag}(1, 2, 4)$ and $\mu = 2.1$, the ratio is $\rho = 0.1/1.1 \approx 0.09$, so each step gains about one decimal digit toward $e_2$. If two distinct eigenvalues are equally close to $\mu$, then $\rho = 1$ and the iteration need not converge to an eigenvector. In practice one factors $A - \mu I = LU$ once and solves a linear system at each step instead of forming the inverse. The near-singularity of $A - \mu I$ for a good shift does not harm the method: the error in the solve is concentrated in the direction of the wanted eigenvector.

Going deeper

For a student

Inverse iteration is the standard way to compute an eigenvector once a good approximation of its eigenvalue is known. Such an approximation may come from the QR algorithm or from bisection on a tridiagonal matrix. Updating the shift at each step with the Rayleigh quotient gives Rayleigh quotient iteration, which converges much faster. The rate of convergence is linear with ratio $\rho$, which is why a shift near the eigenvalue matters.