Numerical Linear Algebra with Applications

5.7 Problems

5.1 Find the eigenvalues and associated eigenvectors for the matrix $\left[\begin{array}{cc}\hfill 1\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 9\hfill \end{array}\right]\text{.}$

5.2 Find the eigenvalues and associated eigenvectors for the matrix $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 6\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill -2\hfill & \hfill -1\hfill \end{array}\right]\text{.}$

5.3 Let $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right]$ Verify that (A^− 1)^T = (A^T)^− 1.

5.4 Let $A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 4\hfill \\ \hfill 2\hfill & \hfill 3\hfill \end{array}\right],B=\left[\begin{array}{cc}\hfill -1\hfill & \hfill 7\hfill \\ \hfill 1\hfill & \hfill -4\hfill \end{array}\right]\text{.}$ Verify that AB and BA have the same eigenvalues.

5.5 Show that the following properties hold for similar ...