\documentclass[fleqn]{article}
%\pagestyle{empty}
\setlength{\topmargin}{0in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{8in}
\setlength{\textwidth}{6in}



\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{enumerate}

\newcommand{\stbar}{\;|\;}
\newcommand{\R}{\mathbb{R}}
\newcommand{\ra}{\rightarrow}
\newcommand{\dotprod}{\bullet}
\newcommand{\proj}{\mathrm{Proj}}
\renewcommand{\span}{\mathrm{Span}}


\theoremstyle{definition}
\newtheorem{theorem}{Theorem}
\newtheorem{example}[theorem]{Example}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{fact}[theorem]{Fact}
\newtheorem{definition}[theorem]{Example}


\newtheorem{exercise}{Exercise}
\newtheorem{maple}{Maple Exercise}



\begin{document}

\begin{center}
\textbf{Worksheet on Eigenvalues and Eigenvectors}
\end{center}


For the matrices below, do the following and answer the questions:
\begin{enumerate}[(a)]
\item  Let $n$ be the number of columns (or rows) of $A$.
\item  Find all eigenvalues of $A$.
\item  For each eigenvalue, find a basis for the associated eigenspace.
  What is the dimension of each eigenspace?
\item  If you throw the bases for all the eigenspaces together, do you get a basis for $\R^n$?
\item  Is it possible to write any vector in $\R^n$ as a combination of eigenvectors?
\item  Is the matrix \emph{deficient}?
\end{enumerate}

\begin{itemize}
\item Some of these are quite easy and were meant to be that way -- they illustrate
  what is supposed to be typical behavior of eigenvalues of a matrix.
\item  For the more difficult ones, you should probably use Maple.
  When using Maple, I think you will understand more if you do it ``by hand,'' 
  meaning that you don't use Maple's powerful eigenvalue functions.
  Use maple to solve equations and row reduce matrices.
  Here are some good Maple commands:
  \begin{itemize}
  \item  \texttt{solve} (this will solve the equations to find eigenvalues)
  \item  \texttt{eigenvalues} (use this to check your work only!)
  \item  \texttt{eigenvects} (use this to check your work only!)
  \end{itemize}
\end{itemize}

\begin{enumerate}
\item  Identity matrix.  What do you think would happen for the $n\times n$ identity matrix?
  \[
  \left[
    \begin{array}{rr}
      1 & 0 \\
      0 & 1
    \end{array}
    \right]
  \]

\item  Zero matrix.   What do you think would happen for the $n\times n$ zero matrix?
  \[
  \left[
    \begin{array}{rr}
      0 & 0 \\
      0 & 0
    \end{array}
    \right]
  \]

\item  A singular matrix.   What do you think would happen for the $n\times n$ singular matrix?
  \[
  \left[
    \begin{array}{rr}
      1 & 2 \\
      2 & 4
    \end{array}
    \right]
  \]

\item  Another singular matrix.  Can you make a generalization?  (What is the rank of this matrix?)
  \[
  \left[
    \begin{array}{rrrr}
      1 & 2 & 3 & 4 \\
      2 & 4 & 6 & 8 \\
      3 & 6 & 9 & 12 \\
      4 & 8 & 12 & 16
    \end{array}
    \right]
  \]

\item  Another singular matrix.  Can you make a generalization?  (What is the rank of this matrix?)
  \[
  \left[
    \begin{array}{rrrr}
      1 & 2 & 3 & 4 \\
      2 & 4 & 6 & 8 \\
      3 & 6 & 9 & 12 \\
      0 & 0 & 1 & -2
    \end{array}
    \right]
  \]

\item  A diagonal matrix.  What do you think would happen for a general $n\times n$ diagonal matrix?
  \[
  \left[
    \begin{array}{rrr}
      2 & 0 & 0 \\
      0 & -3 & 0 \\
      0 & 0 & 5
    \end{array}
    \right]
  \]

\item  This is an interesting one, why?
  It is triangular, so we can immediatly see the eigenvalues.
  But, what about the eigenvectors?
  \[
  \left[
    \begin{array}{rr}
      2 & 1 \\
      0 & 2
    \end{array}
    \right]
  \]

\item  Why is this different from the previous matrix?
  \[
  \left[
    \begin{array}{rr}
      2 & 1 \\
      0 & 3
    \end{array}
    \right]
  \]

\item  Similar to the above ones.
  \[
  \left[
    \begin{array}{rrr}
      2 & 1 & 0 \\
      0 & 2 & 0 \\
      0 & 0 & 3
    \end{array}
    \right]
  \]

\item  
  \[
  \left[
    \begin{array}{rrr}
      2 & 1 & 0 \\
      0 & -4 & 0 \\
      0 & 0 & 3
    \end{array}
    \right]
  \]

\item  
  \[
  \left[
    \begin{array}{rrr}
      -4 & 0 & -4 \\
      7 & 3 & 4 \\
      12 & 0 & 10 
    \end{array}
    \right]
  \]

\item  
  \[
  \left[
    \begin{array}{rrr}
      3 & 12 & 24 \\
      -6 & 21 & 12 \\
      12 & -12 & -9
    \end{array}
    \right]
  \]

\item  
  \[
  \left[
    \begin{array}{rrr}
      3 & 1 & 0 \\
      0 & 3 & 1 \\
      0 & 0 & 3
    \end{array}
    \right]
  \]

\item  
  \[
  \left[
    \begin{array}{rrr}
      14 & 3 & \frac{14}{3} \\
      -3 & -14 & \frac{2}{3} \\
      1 & 2 & 17
    \end{array}
    \right]
  \]


\end{enumerate}



\end{document}






