Chapter 9.1, Problem 41E

Consider a noninvertible $2\times 2$ matrix A with two distinct eigenvalues. (Note that one of the eigenvalues must be 0.) Choose two eigenvectors ${\stackrel{\to }{v}}_1$ and ${\stackrel{\to }{v}}_2$ with eigenvalues ${\lambda }_1=0$ and ${\lambda }_2$ as shown in the accompanying figure. Suppose ${\lambda }_2$ is negative. Sketch a phase portrait for the system $d\stackrel{\to }{x}/dt=A\stackrel{\to }{x}$ , clearly indicating the shape and long−term behavior of the trajectories.