Convergence of diagonalizable fixed-shift inverse iteration ($q_{\mu }=|{\lambda }_{\ast }-\mu |/{\min }_{{\lambda }_j\ne {\lambda }_{\ast }}|{\lambda }_j-\mu |$)

Suppose a diagonalizable matrix has a simple eigenvalue ${\lambda }_{\ast }$ uniquely closest to a shift $\mu \notin \mathrm{spec}(A)$. If the starting vector has a nonzero coefficient $c_{\ast }$ 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 $c_{\ast }({\lambda }_{\ast }-\mu {)}^{-k}$ leaves the target eigenvector plusThis 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, ${\lambda }_{\ast }<\mu$ 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.