Exercise1

For each of the \(2\times2\) matrices below, find the eigenvalues and, when appropriate, the eigenvectors to classify the dynamical system \(\xvec_{k+1}=A\xvec_k\text{.}\) Use this information to sketch the phase portraits.

  1. \(A = \left[\begin{array}{rr} 3\amp 1 \\ 1 \amp 3 \\ \end{array}\right]\text{.}\)

  2. \(A = \left[\begin{array}{rr} 3\amp -2 \\ 4 \amp -1 \\ \end{array}\right]\text{.}\)

  3. \(A = \left[\begin{array}{rr} 1.9 \amp 1.4 \\ -0.7 \amp -0.2 \\ \end{array}\right]\text{.}\)

  4. \(A = \left[\begin{array}{rr} 1.1 \amp -0.2 \\ 0.4 \amp 0.5 \\ \end{array}\right]\text{.}\)

in-context