I have a question about the simultaneous iteration. I am currently working for an exam and I do not understand this step (taken from Numerical Linear Algebra from Trefethen/Bau):
For the power iteration it holds, that for an arbitrary starting vector $v^{(0)}$ with $||v^{(0)}|| = 1$ that $A^kv \rightarrow q_j$ for $k \rightarrow \infty$, where $q_j$ is the eigenvector corresponding to the maximum eigenvalue (in absolute value). So far, so good.
Now the book states the following: For a set of n linearly independent vectors $v_1^{(0)}, ..., v_n^{(0)}$, the space $span\{A^kv_1^{(0)}, ..., A^kv_n^{(0)}\}$ converges to the space $span\{q_1, ..., q_n\}$, i.e. the space spanned by the eigenvectors. And this is something I do not understand. Above it is stated, that for almost any arbitrary starting vector, $v$ would converge to the eigenvector corresponding to the largest eigenvalue. How can than the space spanned by several vectors span the same space as all eigenvectors? I would assume that the all will converge sooner or later to the same vector, thus only spanning a one-dimensional subspace.
Where is my mistake?