% LaTeX file -- M245, Fall 2012, Assignment 2

\documentclass[11pt]{amsart}
\usepackage{amssymb, amstext, amscd, amsmath}
\pagestyle{empty}
%\topmargin 1cm
%\textheight 9in
\textwidth 5.5in
\oddsidemargin .5in
\evensidemargin .5in
%\voffset -1.5cm

\newcommand{\bF}{{\mathbb F}}
\newcommand{\bN}{{\mathbb N}}
\newcommand{\bQ}{{\mathbb Q}}
\newcommand{\bR}{{\mathbb R}}
\newcommand{\bC}{{\mathbb C}}
\newcommand{\bZ}{{\mathbb Z}}

%      Capital script letters
  \newcommand{\A}{{\mathcal{A}}}
  \newcommand{\B}{{\mathcal{B}}}
  \newcommand{\C}{{\mathcal{C}}}
  \newcommand{\D}{{\mathcal{D}}}
  \newcommand{\E}{{\mathcal{E}}}
  \newcommand{\F}{{\mathcal{F}}}
  \newcommand{\G}{{\mathcal{G}}}
\renewcommand{\H}{{\mathcal{H}}}
  \newcommand{\I}{{\mathcal{I}}}
  \newcommand{\J}{{\mathcal{J}}}
  \newcommand{\K}{{\mathcal{K}}}  
\renewcommand{\L}{{\mathcal{L}}}
  \newcommand{\M}{{\mathcal{M}}}
  \newcommand{\N}{{\mathcal{N}}}
\renewcommand{\O}{{\mathcal{O}}}
\renewcommand{\P}{{\mathcal{P}}}
  \newcommand{\Q}{{\mathcal{Q}}}
  \newcommand{\R}{{\mathcal{R}}}
\renewcommand{\S}{{\mathcal{S}}}
  \newcommand{\T}{{\mathcal{T}}}
  \newcommand{\U}{{\mathcal{U}}}
  \newcommand{\V}{{\mathcal{V}}}
  \newcommand{\W}{{\mathcal{W}}}
  \newcommand{\X}{{\mathcal{X}}}
  \newcommand{\Y}{{\mathcal{Y}}}
  \newcommand{\Z}{{\mathcal{Z}}}

\newcommand{\FOR}{\text{ for }}
\newcommand{\FORAL}{\text{ for all }}
\newcommand{\AND}{\text{ and }}
\newcommand{\IF}{\text{ if }}
\newcommand{\OR}{\text{ or }}
\newcommand{\qand}{\quad\text{and}\quad}
\newcommand{\qor}{\quad\text{or}\quad}
\newcommand{\qfor}{\quad\text{for}\quad}
\newcommand{\qforal}{\quad\text{for all}\quad}
\newcommand{\qif}{\quad\text{if}\quad}

\newcommand{\ol}{\overline}
\newcommand{\ep}{\varepsilon}
\newcommand{\dlim}{\displaystyle\lim\limits}
\renewcommand{\phi}{\varphi}
\newcommand{\mt}{\emptyset}
\newcommand{\diag}{\operatorname{diag}}
\newcommand{\ran}{\operatorname{Ran}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\spn}{\operatorname{span}}
\newcommand{\Tr}{\operatorname{Tr}}

\newcounter{exerci}[section]
\renewcommand{\theexerci}{\arabic{exerci}.}
\newenvironment{Ex}{\begin{list}%
 {\theexerci\hfill}{\usecounter{exerci}\rightmargin=0pt\leftmargin=12pt%
 \labelwidth=12pt\labelsep=6pt\itemsep=3ex}}{\end{list}\medbreak}
\newenvironment{parts}{\vspace{1ex}\begin{enumerate}\renewcommand{\itemsep}{1ex}
 \renewcommand{\labelenumi}{(\alph{enumi})}}{\end{enumerate}}
\newcommand{\hint}{\textbf{Hint:} }

\begin{document}
\thispagestyle{empty}

%%%%%%%   TITLE   %%%%%%
\begin{center}
\textbf{\large Math 245 \\ Assignment 2\\
Due Friday October 5}
\end{center}
\vskip 2em 


\begin{Ex}

\item Consider the linear recursion
$a_{n+3} = 3a_{n+2}-a_{n+1}-a_n$ for $n \ge 0$.
\begin{parts}
\item If  $a_0=a_1=1$ and $a_2=2$, find a formula for $a_n$.
\item Find $\dlim_{n\to\infty} a_n^{1/n}$.\\ 
What happens if $a_0=a_1=a_2=1$?\\
What happens if $a_0=a_1=1$ and $a_2=1+\ep$ for some very small $\ep \ne 0$?
\end{parts}

\item  Let $E \in \L(V)$ such that $E$ is idempotent, i.e.\ $E^2=E$.
\begin{parts}
\item Let $R = \ran E$ and $K = \ker E$. Show that $V = R \oplus K$.
\item Express $E$ as a $2\times2$ matrix with \textit{explicit} matrix entries with respect
to this decomposition.
\item Show that $E$ is diagonalizable.
\item Suppose that $v_1,v_2,v_3,v_4$ is a basis for $V$ and $R=\spn\{v_1,v_2\}$.
Find the matrices of \textit{all} idempotents $E$ with range $R$ w.r.t.\ this basis.
\end{parts}


\item 
\begin{parts}
\item Suppose that $A \in \L(V)$ has $n=\dim V$ distinct eigenvalues.
Suppose that $B\in\L(V)$ commutes with $A$ (i.e. $AB=BA$).
Show that there is a polynomial $p$ so that $B=p(A)$.
\item Show that this conclusion fails for the matrix
\[ A = \begin{bmatrix}0&1&0&0\\0&0&0&0\\0&0&0&1\\0&0&0&0\end{bmatrix} .\]
\end{parts}

\item Let $V$ be a finite dimensional vector space over $\bC$.
Suppose that $A,B \in \L(V)$ commute (i.e. $AB=BA$).
\begin{parts}
\item  Show that if $p \in \bC[x]$ is any polynomial, then $\ker(p(A))$ is invariant for $B$.
\item Show that $A$ and $B$ have a common eigenvector.\\
\hint Take $\alpha\in\sigma(A)$ and let $B_1$ be the restriction of $B$ to $W=\ker(A-\alpha I)$.
Find an eigenvector for $B_1$.
\item Show that $A$ and $B$ can be simultaneously triangularized.  i.e. find a basis $\B$
so that both $\big[A\big]_\B$ and $\big[B\big]_\B$ are triangular.
\hint use induction on $n$ as in the proof of triangularizability of a single map.

\item \textbf{Bonus.}  Generalize this to a commuting family of transformations on $V$.
\end{parts}




\end{Ex}
\end{document}