Chapter 9.1, Problem 19P

(a)

To determine

To show: The eigenvalues of the system are $r_1=0.5+1.5i$, and $r_1=0.5-1.5i$.

(b)

To determine

To show: That the eigen vector ${\xi }^{(1)}=\left(\begin{array}{c}5\\ 3-3i\end{array}\right)$.

(c)

To determine

To show: That $y^{\prime }=\left(P^{-1}AP\right)y$, where P is $\left(\begin{array}{cc}5& 0\\ 3& -3\end{array}\right)$.

(d)

To determine

The value of $P^{-1}$ and proved that $\left(P^{-1}AP\right)=\left(\begin{array}{cc}0.5& 1.5\\ -1.5& 0.5\end{array}\right)$, where P is $\left(\begin{array}{cc}5& 0\\ 3& -3\end{array}\right)$.