= Solution
{c}
Choose a shift $\mu$ which is not an <eigenvalue>. <Inverse iteration> applies <power iteration> to $B=(A-\mu I)^{-1}$, implemented by a <linear system> solve rather than by forming an explicit <matrix inverse>:
$$
\boxed{(A-\mu I)w_{k+1}=v_k,\qquad
v_{k+1}=\frac{w_{k+1}}{\|w_{k+1}\|_2}.}
$$
A single <LU decomposition> of $A-\mu I$ can be reused for all the solves at this fixed shift. An <eigenpair> $Au_j=\lambda_ju_j$ transforms into
$$
Bu_j=\frac1{\lambda_j-\mu}u_j.
$$
Thus an <eigenvalue> $\lambda_*$ uniquely closest to $\mu$ becomes the <dominant eigenvalue> of $B$. For a <diagonalizable matrix>, the argument in the preceding subpart gives <convergence of diagonalizable fixed-shift inverse iteration>, provided the starting coefficient in the target <eigenspace> is nonzero, with <power iteration direction error> $O(q_\mu^k)$, where
$$
\boxed{q_\mu=
\frac{|\lambda_*-\mu|}{\min_{j:\lambda_j\ne\lambda_*}|\lambda_j-\mu|}<1.}
$$
For any prescribed <simple eigenvalue>, taking a sufficiently close but unequal shift makes the numerator small while the denominator stays close to its positive separation from the remaining <eigenvalues>. This makes $q_\mu$ small and gives very fast convergence per iteration. The <Rayleigh quotient> of $v_k$ estimates $\lambda_*$ directly. If $\beta_k$ estimates the associated <eigenvalue> of $B$, the corresponding estimate is $\mu+\beta_k^{-1}$.
The selection principle also applies when the other <eigenvalues> have nontrivial blocks in the <Jordan normal form>: their contributions acquire powers of $k$, multiplying strictly smaller geometric factors. A simple uniquely closest <eigenvalue> still attracts the direction when the <Riesz projection> of $v_0$ onto its generalized <eigenspace> is nonzero. For nonreal target <eigenvalues>, allow complex shifts and arithmetic; a real shift cannot distinguish a conjugate pair of equal distances. Setting $\mu=\lambda_*$ makes the solve singular, and an exactly zero starting target component prevents convergence to that target in exact arithmetic. When $\lambda_*<\mu$ is real, account for the <sign behavior of fixed-shift inverse iteration>.
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