Let $A \in \mathbb{R}^{n \times n}$ be a real matrix with $n$ linearly independent eigenvectors. The eigenvalues of $A$ can be calculated from the sequence $x^{(k)}, k=0,1, \ldots$, which is generated by the power method

$$x^{(k+1)}=\frac{A x^{(k)}}{\left\|A x^{(k)}\right\|},$$

where $x^{(0)}$ is a real nonzero vector.

(i) Describe the asymptotic properties of the sequence $x^{(k)}$ in the case that the eigenvalues $\lambda_{i}$ of $A$ satisfy $\left|\lambda_{i}\right|<\left|\lambda_{n}\right|, i=1, \ldots, n-1$, and the eigenvectors are of unit length.

(ii) Present the implementation details for the power method for the setting in (i) and define the Rayleigh quotient.

(iii) Let $A$ be the $3 \times 3$ matrix

$$A=\lambda I+P, \quad P=\left(\begin{array}{lll} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right)$$

where $\lambda$ is real and nonzero. Find an explicit expression for $A^{k}, k=1,2,3, \ldots$

Let the sequence $x^{(k)}$ be generated by the power method as above. Deduce from your expression for $A^{k}$ that the first and second components of $x^{(k+1)}$ tend to zero as $k \rightarrow \infty$. Further show that this implies $A x^{(k+1)}-\lambda x^{(k+1)} \rightarrow 0$ as $k \rightarrow \infty$.