Simultaneous iteration interpretation of the QR algorithm

The accumulated orthogonal factor in QR iteration is the $Q$ factor of a matrix power, so its leading columns span the same spaces as simultaneous power iteration.

Let a real symmetric matrix have an orthonormal eigenbasis $w_1,\dots ,w_n$, withWrite two starting vectors asFor two-column subspace iteration to recover
$\mathrm{span}(w_{n-1},w_n)$, the two projections onto that dominant subspace must be independent. The exact condition isRequiring each of the four coefficients to be nonzero does not imply this determinant condition.