= Convergence of diagonalizable fixed-shift inverse iteration
{title2=$q_\mu=|\lambda_*-\mu|/\min_{\lambda_j\ne\lambda_*}|\lambda_j-\mu|$}
Suppose a <diagonalizable matrix> has a simple <eigenvalue> $\lambda_*$ uniquely closest to a shift $\mu\notin\operatorname{spec}(A)$. If the starting <vector> has a nonzero coefficient $c_*$ in the target <eigenbasis> direction, fixed-shift <inverse iteration> converges to that <eigenvector> line. In fact, the <eigenvalues> of $(A-\mu I)^{-1}$ are $(\lambda_j-\mu)^{-1}$, so after $k$ inverse applications, division by $c_*(\lambda_*-\mu)^{-k}$ leaves the target <eigenvector> plus
$$
\sum_{j\ne *}\frac{c_j}{c_*}
\left(\frac{\lambda_*-\mu}{\lambda_j-\mu}\right)^ku_j.
$$
This proves direction error $O(q_\mu^k)$ in <Big O notation>. The multiplicative constant depends on the <eigenbasis> and starting coefficients, and no uniform reduction at the first step follows for a <nonsymmetric matrix>. Over the real numbers, $\lambda_*<\mu$ produces alternating signs. A nonzero ordinary <inner product> with a right <eigenvector> is not generally the same as a nonzero coefficient in the <eigenbasis>; the latter is the required condition.
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