Chapter 7.8, Problem 19P

(a)

To determine

To show: The coefficient matrix A has triple eigen value $r=1$ and there are exists only two linearly independent eigen vectors ${\xi }^{\left(1\right)}=\left(\begin{array}{c}1\\ 0\\ 2\end{array}\right),{\xi }^{\left(2\right)}=\left(\begin{array}{c}0\\ 2\\ -3\end{array}\right)$.

(b)

To determine

To show: The vectors $\xi$ and $\eta$ must satisfy the equations $\left(A-I\right)\xi =0\text{ and }\left(A-I\right)\eta =\xi$.

(c)

To determine

To show: The vector $\eta$ satisfies the equation ${\left(A-I\right)}^2\eta =0$ and further show that ${\left(A-I\right)}^2=0$.

(d)

To determine

The corresponding vector $\xi$ when $\eta$ is chosen as $\eta ={\left(0,0,1\right)}^T$.

(e)

To determine

A fundamental matrix for the given system, $x^{\prime }=\left(\begin{array}{ccc}5& -3& -2\\ 8& -5& -4\\ -4& 3& 3\end{array}\right)x$.

(f)

To determine

The matrix $T^{-1}$ and form the product $J=T^{-1}AT$.