Exercise6

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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