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\centerline{Mathematics 244 -- Linear Algebra}
\centerline{Discussion 3 -- Discrete Dynamical Systems}
\centerline{April 16, 2007}
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\noindent
{\it Background}
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The eigenvalues and eigenvectors of the matrix $A$ can be 
used to study the long-term behavior of discrete-time
dynamical systems of the form
$$x_{k+1} = A x_k.\leqno(1)$$
Here $x_k \in {\bf R}^n$ represents the state of the system 
at time $t = k$, $A \in M_{n\times n}({\bf R})$ is the ``stage 
matrix'' that connects the state of the system at time $t = k$ 
to the state at time $t = k + 1$.  Note that the stage matrix
$A$ is a matrix of constants -- it does not depend on the time.
It would also be possible to consider more general systems 
$$x_{k+1} = A_k x_k,$$
in which the way things are changing also changes from time to time.
But we will stick to systems of this form (1).  

(Comment:  A special class, in which the sum of the entries in 
each column of $A$ is equal to $1$, are 
often called {\it Markov chains}.  Tanya Leise showed in her
colloquium talk on Google that he stage matrix $A$ in a Markov
chain always has $\lambda = 1$ as an eigenvalue.  This fact
was a key step in the analysis of the Google
``importance vector.''  Also see section 4.9 in our text for
more about this special case.)

In the discussion questions I,II,III below, you will study the 
systems with $n = 2$ of the form
$$x_{k+1} = \pmatrix{.4 & .3 \cr
                     -p & 1.2\cr} x_k\leqno(2)$$
for various values of $p$.  Here $x_k = \pmatrix{O_k \cr S_k\cr}$
derives from our in-class example of the interactions of 
predators (spotted owls) and prey (squirrels).  The components of $x_k$
represent the populations of owls and squirrels at time $t = k$.  
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\noindent
{\it Discussion Questions}
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\noindent
I.  Suppose $p = .5$.  
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\item{A)}  What are the eigenvalues and eigenvectors of $A$ (decimal 
approximations are OK here).
Give a basis of ${\bf R}^2$ consisting of eigenvectors.
\item{B)}  Suppose the initial populations are 
$x_0 = \pmatrix{O_0 \cr S_0\cr} = \pmatrix{10 \cr 30\cr}.$
Express $x_0$ as a linear combination of the eigenvector basis you found
in part A, and give a formula for $x_k$ for general $k$ based on
this.  What happens to the populations ``in the long run?''
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\noindent
II.  
\item{A)}  What is the {\it largest} value of $p$ for which the matrix
$A$ has two {\it real} eigenvalues?  
\item{B)}  Say $p = 1$ now.  
The method we used in question I breaks down if there are
no real eigenvectors.  However, we can still compute $x_k$, plot them,
and try to understand the behavior geometrically.  Starting from 
$x_0 = \pmatrix{10 \cr 30\cr}$ again, compute $x_1, x_2, \ldots, x_{10}$
and plot them carefully (e.g. on graph paper or using Maple).  What is 
your ``best guess'' about the long-term behavior here?
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\noindent
III.  The system (2) is only realistic as a mathematical model
of a predator-prey system when $p > 0$.  (This is because 
the presence of the predators should have a {\it negative effect} on
the population of the prey -- they are getting eaten, after all!)
Nevertheless, we can still look at dynamical systems of the form
(2) from the mathematical point of view for all real values of $p$.
And the systems with $p \le 0$ could model other types of real-world
systems. 
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\item{A)}  What are the eigenvalues of $A$ when $p = -1.6$?  Find
an eigenvector for each, and give a formula for $x_k$ based on
an expansion of $x_0$ in terms of the eigenvalues (like what you did in part 
I B).
\item{B)}  Same question as in part A, but taking $p = -2$.  
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IV.  For a general $2 \times 2$ matrix 
$A = \pmatrix{a & b\cr c & d}$, the scalar $a + d$ is called the {\it trace}
of $A$, writen ${\rm Tr}(A)$.  The scalar $ad - bc$ is $\det(A)$.
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\item{A)} Write the characteristic polynomial of $A$ using ${\rm Tr}(A)$ and
$\det(A)$ to express the coefficients.  
\item{B)}  Using the characteristic polynomial of $A$:
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\itemitem{1)}  When are both eigenvalues of $A$ real and distinct and of 
the same sign? 
\itemitem{2)}  When are both eigenvalues of $A$ real and distinct but of 
opposite signs?  
\itemitem{3)}  When does $A$ have a repeated (real) eigenvalue?  
\itemitem{4)}  When does $\det(A - \lambda I) = 0$ have no real roots (that is,
$A$ has no real eigenvalues)? 
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\item{} Express your answers using equalities/inequalities involving
$\det(A)$ and ${\rm Tr}(A)$.  Hint: Think quadratic formula(!)
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\item{C)}  Each of the conditions from part B corresponds to a subset
of the $({\rm Tr},\det)$-plane, where we can ``plot a matrix'' $A$ according
to the values of its trace and determinant.  (For example, 
the matrix $A = \pmatrix{3 & -2 \cr 2 & -4\cr}$ has ${\rm Tr}(A) = 3 - 4 = -1$
and $\det(A) = -8$. We would represent it as the point $(-1,-8)$ in the 
$({\rm Tr},\det)$ plane.)  Draw and shade the regions
corresponding to each condition from part B.
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\noindent
{\it Assignment}
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Group write-ups due:  Monday, April 23.
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