Exercise5

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 $$\xvec_{k+1} = A\xvec_k\text{.}$$

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 $$\xvec_{k+1} = A^{-1}\xvec_k\text{.}$$

4. If a $$4\times4$$ matrix has complex eigenvalues $$\lambda_1\text{,}$$ $$\lambda_2\text{,}$$ $$\lambda_3\text{,}$$ and $$\lambda_4\text{,}$$ all of which satisfy $$|\lambda_j| \gt 1\text{,}$$ 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.

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