Orthogonal iteration

This process can be simplified into a single kind of iteration: the orthogonal iteration.

$$Z=A{Q}_k,Q_{k+1}{R}_{k+1}=Z$$

Column $i+1$ of $Q_{k+1}$ converges to column $i+1$ of $P_{\{q_1,q_2,\dots ,q_i{\}}^{\perp }}A{Q}_k$.

In fact, $Q_k\to Q$.

Note: in the Schur decomposition, $Q$ is not unique. We have $QT{Q}^H$. We can multiply $Q$ by any diagonal matrix $Q{e}^{i\Theta }$ with $\Theta$ a diagonal real matrix. This does not change the diagonal of $T$ or its upper triangular structure. It does change the strict upper diagonal entries, though.

So $Q_k\to Q$ up to some diagonal matrix $e^{i{\Theta }_k}$. Or more simply

$$\text{span}\{[Q_k{]}_{,j}\}\to \text{span}\{q_{,j}\}$$

The angle between these subspaces goes to 0.