Sign behavior of fixed-shift inverse iteration

If the closest eigenvalue ${\lambda }_{\ast }$ lies above the shift, the dominant eigenvalue $({\lambda }_{\ast }-s{)}^{-1}$ of the inverse is positive and normalized iterates approach one fixed sign of its eigenvector. If ${\lambda }_{\ast }<s$, that inverse eigenvalue is negative, so the direction converges but consecutive normalized vectors alternate signs. At a shift exactly midway between two simple eigenvalues, the two inverse eigenvalues have equal modulus and opposite signs, so generic iterates alternate between two mixtures instead of selecting one eigenvector.