###### Exercise 2

We will consider matrices that have the form \(A=PDP^{-1}\) where

\begin{equation*}
D = \mattwo p00{\frac12},
P = \mattwo 2{-2}11
\end{equation*}

where \(p\) is a parameter that we will vary. Sketch phase portraits for \(D\) and \(A\) below when

\(p=\frac12\text{.}\)

\(p=1\text{.}\)

\(p=2\text{.}\)

For the different values of \(p\text{,}\) determine which types of dynamical system results. For what range of \(p\) values do we have an attractor? For what range of \(p\) values do we have a saddle? For what value does the transition between the two types occur?