Power iteration If $A$ is a matrix you can calculate its largest eigenvalue $\lambda_1$. What are the exact conditions under which the power iteration converges? Power iteration
Especially, I often see that we demand that the matrix is symmetric? Is this necessary?
What seems to be indespensable is that there is a largest eigenvalue (absolute value is large). But what about the structure of the eigenspace of this eigenvalue?
Apparently, many times it is not considered that the eigenspace to this largest eigenvalue does not need to be one-dimensional. So what happens if the eigenspace is two-dimensional. Can we still use this algorithm?

Let $$M:=\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix},\ \text{then}\quad M \begin{pmatrix}1\\ 1 \end{pmatrix}=1\cdot \begin{pmatrix}1\\ 1 \end{pmatrix},\ \text{and} \quad M \begin{pmatrix}1\\ -1 \end{pmatrix}=-1\cdot \begin{pmatrix}1\\ -1 \end{pmatrix} $$
Now let $x^0 =(x^0_1,x^0_2) \neq (0,0)$, then the sequence $x^{k+1} = Mx^k$ will never converge since $M$ just switch the coordinates of $x^k$. If you want a matrix with maximal eigenvalue (without absolute value) is not simple and the algorithm don't converge, consider $$A = \begin{pmatrix}0 & 1 & 0 \\ 1 &0 & 0 \\ 0& 0 &1\end{pmatrix}, \quad A\begin{pmatrix}1 \\ 1 \\ 0 \end{pmatrix}=1\cdot \begin{pmatrix}1 \\ 1 \\ 0 \end{pmatrix},\quad A\begin{pmatrix}1 \\ -1 \\ 0 \end{pmatrix}=-1\cdot \begin{pmatrix}1 \\ -1 \\ 0 \end{pmatrix},\quad A\begin{pmatrix}0 \\ 0 \\ 1 \end{pmatrix}=1\cdot \begin{pmatrix}0 \\ 0\\ 1 \end{pmatrix}, $$
and the sequence $x^{k+1} = Ax^k$ will never converge except if you start directly with a eigenvector.
