##### Activity4.4.3

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 create a sketch similar to that of Figure 1; that is, on the left, sketch the trajectories $$\coords{A^k\xvec_0}{\bcal}$$ for some initial vectors $$\xvec_0$$ and on the right, sketch $$A^k\xvec_0\text{.}$$ Describe the behavior of $$A^k\xvec_0$$ as $$k$$ becomes very large for a typical initial vector $$\xvec_0\text{.}$$

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

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

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

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

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

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

in-context