Activity 4.4.3
We will now look at several more examples of dynamical systems. If \(P = \left[\begin{array}{rr} 1 \amp 1 \\ 1 \amp 1 \\ \end{array}\right] \text{,}\) we note that the columns of \(P\) form a basis \(\bcal\) of \(\real^2\text{.}\) Given below are several matrices \(A\) written in the form \(A=PEP^{1}\) for some matrix \(E\text{.}\) For each matrix, state the eigenvalues of \(A\) and sketch a phase portrait for the matrix \(E\) on the left and a phase portrait for \(A\) on the right. Describe the behavior of \(A^k\xvec_0\) as \(k\) becomes very large for a typical initial vector \(\xvec_0\text{.}\)

\(A=PEP^{1}\) where \(E = \left[\begin{array}{rr} 1.3 \amp 0 \\ 0 \amp 1.5 \\ \end{array}\right] \text{.}\)

\(A=PEP^{1}\) where \(E = \left[\begin{array}{rr} 0 \amp 1 \\ 1 \amp 0 \\ \end{array}\right] \text{.}\)

\(A=PEP^{1}\) where \(E = \left[\begin{array}{rr} 0.7 \amp 0 \\ 0 \amp 1.5 \\ \end{array}\right] \text{.}\)

\(A=PEP^{1}\) where \(E = \left[\begin{array}{rr} 0.3 \amp 0 \\ 0 \amp 0.7 \\ \end{array}\right] \text{.}\)

\(A=PEP^{1}\) where \(E = \left[\begin{array}{rr} 1 \amp 0.9 \\ 0.9 \amp 1 \\ \end{array}\right] \text{.}\)

\(A=PEP^{1}\) where \(E = \left[\begin{array}{rr} 0.6 \amp 0.2 \\ 0.2 \amp 0.6 \\ \end{array}\right] \text{.}\)