Linear Algebra with Applications, 10th Edition

Section 7.6 Exercises

1. Let

   $$A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$$

   1. Apply one iteration of the power method to A with any nonzero starting vector.

   2. Apply one iteration of the QR algorithm to A.

   3. Determine the exact eigenvalues of A by solving the characteristic equation, and determine the eigen-space corresponding to the largest eigenvalue. Compare your answers with those to parts (a) and (b).

2. Let

   $$\begin{array}{ccc}A=\left[\begin{array}{ccc}2& 1& 0\\ 1& 3& 1\\ 0& 1& 2\end{array}\right]& \text{and}& {\mathbf{u}}_0=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]\end{array}$$

   1. Apply the power method to A to compute ${\mathbf{v}}_1,{\mathbf{u}}_1,{\mathbf{v}}_2,{\mathbf{u}}_2$, and ${\mathbf{v}}_3$. (Round off to two decimal places.)

   2. Determine an approximation ${\lambda }_1^{\prime }$ to the largest eigenvalue of A from the coordinates of ${\mathbf{v}}_3$. Determine the exact value of ${\lambda }_1$ and compare it with ${\lambda }_1^{\prime }$. What is the relative error?

3. Let

   $$\begin{array}{ccc}A=\left[\begin{array}{cc}\hfill 1& \hfill 2\\ \hfill -1& \hfill -1\end{array}\right]& \text{and}& {\mathbf{u}}_0=\left[\begin{array}{c}1\\ 1\end{array}\right]\end{array}$$

   1. Compute ${\mathbf{u}}_1,{\mathbf{u}}_{2,}{\mathbf{u}}_3$, and ${\mathbf{u}}_4$, using the power method.

   2. Explain ...