Let $A \in \mathbb{R}^{n \times n}$ be a real matrix with $n$ linearly independent eigenvectors. When calculating eigenvalues of $A$, the sequence $\mathbf{x}^{(k)}, k=0,1,2, \ldots$, is generated by power method $\mathbf{x}^{(k+1)}=A \mathbf{x}^{(k)} /\left\|A \mathbf{x}^{(k)}\right\|$, where $\mathbf{x}^{(0)}$ is a real nonzero vector.

(a) Describe the asymptotic properties of the sequence $\mathbf{x}^{(k)}$, both in the case where the eigenvalues $\lambda_{i}$ of $A$ satisfy $\left|\lambda_{i}\right|<\left|\lambda_{n}\right|, i=1, \ldots, n-1$, and in the case where $\left|\lambda_{i}\right|<\left|\lambda_{n-1}\right|=\left|\lambda_{n}\right|, i=1, \ldots, n-2$. In the latter case explain how the (possibly complexvalued) eigenvalues $\lambda_{n-1}, \lambda_{n}$ and their corresponding eigenvectors can be determined.

(b) Let $n=3$, and suppose that, for a large $k$, we obtain the vectors

$$\mathbf{y}_{k}=\mathbf{x}_{k}=\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right], \quad \mathbf{y}_{k+1}=A \mathbf{x}_{k}=\left[\begin{array}{l} 2 \\ 3 \\ 4 \end{array}\right], \quad \mathbf{y}_{k+2}=A^{2} \mathbf{x}_{k}=\left[\begin{array}{l} 2 \\ 4 \\ 6 \end{array}\right]$$

Find two eigenvalues of $A$ and their corresponding eigenvectors.