Dominant eigenpair extraction from a two-step Krylov recurrence ($y_{k+2}=s{y}_{k+1}-p{y}_k$)

When two independent iterates span an invariant two-dimensional subspace, their recurrence gives the characteristic polynomial ${\lambda }^2-s\lambda +p$. For distinct roots ${\lambda }_1,{\lambda }_2$, corresponding eigenvectors are $y_{k+1}-{\lambda }_2{y}_k$ and $y_{k+1}-{\lambda }_1{y}_k$. For a merely approximately dominant subspace, the same computation gives approximate eigenpairs. A single orbit cannot separate a repeated eigenvalue's entire eigenspace when its projection is only one direction.