Chapter 9.5, Problem 18E

Consider the system ${\text{x}}^{\prime }\left(t\right)=\text{Ax}\left(t\right),t\ge 0$ with

$\text{A}=\left[\begin{array}{cc}-2& 1\\ 1& -2\end{array}\right]$

a. Show that the matrix A has eigenvalues $r_1=-1\text{ and }{r}_2=-3$ with corresponding eigenvectors $u_1=\text{col}\left(1,1\right)\text{ and }{u}_2=\text{col}\left(1,-1\right)$.

b. Sketch the trajectory of the solution having initial vector $\text{x}\left(0\right)={\text{u}}_1$.

c. Sketch the trajectory of the solution having initial vector $\text{x}\left(0\right)=-{\text{u}}_2$.

d. Sketch the trajectory of the solution having initial vector $\text{x}\left(0\right)={\text{u}}_1-{\text{u}}_2$.