Chapter 5.8, Problem 6E

Let A = $\left[\begin{array}{cc}-2& -3\\ 6& 7\end{array}\right]$. Repeat Exercise 5, using the following sequence x, Ax, …, A⁵x.

$\left[\begin{array}{l}1\\ 1\end{array}\right],\left[\begin{array}{l}-5\\ 13\end{array}\right],\left[\begin{array}{l}-29\\ 61\end{array}\right],\left[\begin{array}{l}-125\\ -253\end{array}\right],\left[\begin{array}{l}-509\\ 1021\end{array}\right],\left[\begin{array}{l}-2045\\ 4093\end{array}\right]$

5. Let A = $\left[\begin{array}{cc}15& 16\\ -20& -21\end{array}\right]$. The vectors x, …, A⁵x are $\left[\begin{array}{l}1\\ 1\end{array}\right],\left[\begin{array}{l}31\\ -41\end{array}\right],\left[\begin{array}{l}-191\\ 241\end{array}\right],\left[\begin{array}{l}991\\ -1241\end{array}\right],\left[\begin{array}{l}-4991\\ 6241\end{array}\right],\left[\begin{array}{l}24991\\ -31241\end{array}\right]$.

Find a vector with a 1 in the second entry that is close to an eigenvector of A. Use four decimal places. Check your estimate, and give an estimate for the dominant eigenvalue of A.