Regarding the text below, I have two questions :
1)
I don't know where $q_1\approx\frac{A^kv^{(0)}}{\parallel A^kv^{(0)} \parallel}$ comes from.
For me, as $k \rightarrow \infty$ we get $q_1\approx \frac{A^kv^{(0)}}{\lambda_1^kc_1}$ and I don't see how the $\lambda_1^kc_1$ is being replaced by $\parallel A^kv^{(0)} \parallel$ in the text.
2)
Regarding the drawbacks of this method, how would it behave (Could it fail?) for a 12x12 symmetric matrix with 100 on the diagonal and reals ranging from 70 to 99 elsewhere ?
Thank you
http://mathreview.uwaterloo.ca/archive/voli/1/panju.pdf Page 3/10
For i). $q_1$ is unitary, then in the form $u/||u||$.
For ii). It's your business to test the considered algorithm and I will not do it. Yet, your matrix is closed to $A=[a_{i,j}]$ with $a_{i,j}\approx 100$; then the eigenvalues are closed to $0$ ($11$ times) and $12\times 100=1200$. Then the quotient $|\lambda_2/\lambda_1|$ is very small and the convergence is very fast.
Note that the power method works for complex matrices under the condition that $|\lambda_1|$ is strictly larger than the other $|(\lambda_i)_{i>1}|$ (which may be equal to each other). The speed of convergence is geometric with coefficient $|\lambda_2|/|\lambda_1|$