Exercise 6

Determine whether the following statements are true or false and provide a justification for your response. In each case, we are considering a dynamical system of the form $\xvec_{k+1} = A\xvec_k$.

1. If the $2\times2$ matrix $A$ has a complex eigenvalue, we cannot make a prediction about the behavior of the trajectories.

2. If $A$ has eigenvalues whose absolute value is smaller than 1, then all the trajectories are pulled in toward the origin.

3. If the origin is a repellor, then it is an attractor for the system $\xvec_{k+1} = A^{-1}\xvec_k$.

4. If a $4\times4$ matrix has complex eigenvalues $\lambda_1$, $\lambda_2$, $\lambda_3$, and $\lambda_4$, all of which satisfy $|\lambda_j| \gt 1$, then all the trajectories are pushed away from the origin.

5. If the origin is a saddle, then all the trajectories are pushed away from the origin.