Chapter 7.5, Problem 34E

Exercise 33 illustrates how you can use the powers of a matrix to find its dominant eigenvalue (i.e., the eigenvalue with maximal modulus), at least when this eigenvalue is real. But what about the other eigenvalues?
a. Consider an $n\times n$ matrix A with n distinct complex eigenvalues ${\lambda }_1,{\lambda }_2,\mathrm{...},{\lambda }_n$ , where ${\lambda }_1$ is real. Suppose you have a good (real) approximation $\lambda$ of ${\lambda }_1$ (good in that $\left|\lambda -{\lambda }_1\right|<\left|\lambda -{\lambda }_i\right|$ , for $i=2,\mathrm{...},n$ ). Consider the matrix $A-\lambda I_n$ . What are its eigenvalues? Which has the smallest modulus? Now consider the matrix ${\left(A-\lambda I_n\right)}^{-1}$ . What are its eigenvalues? Which has the largest modulus? What is the relationship between the eigenvectors of A and those of ${\left(A-\lambda I_n\right)}^{-1}$ ? Consider higher and higher powers of ${\left(A-\lambda I_n\right)}^{-1}$ . How does this help you lo find an eigenvector of A with eigenvalue ${\lambda }_1$ , and ${\lambda }_1$ itself? Use the results of Exercise 33.
b. As an example of part a, consider the matrix
$A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 10\end{array}\right]$ .
We wish to find the eigenvectors and eigenvalues of A without using the corresponding commands on the computer (which is. after all, a “black box”). First, we find approximations for the eigenvalues by graphing the characteristic polynomial (use technology). Approximate the three real eigenvalues of A to the nearest integer. One of the three eigenvalues of A is negative. Find a good approximation for this eigenvalue and a corresponding eigenvector by using the procedure outlined in part a. You are not asked to do the same for the two other eigenvalues.