\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig,amsmath,latexsym,bm}
\usepackage{enumerate}
\usepackage{tikz}
\usepackage[all]{xy}
\usepackage[numbers,square,sort&compress]{natbib}
\usepackage{bibentry}
\usepackage{stackrel}
\usepackage[]{hyperref}
\usetikzlibrary{arrows}
\usetikzlibrary{decorations.markings}
\usepackage{url}

\tikzset{big arrow/.style={decoration={markings,mark=at position 0.6
      with {\arrow[scale=2]{>}}}, postaction={decorate}, shorten >=1pt}}

% Code to make an augmented matrix (Usage: \begin{bmatrix}[ccc|c])
\makeatletter
\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{%
  \hskip -\arraycolsep
  \let\@ifnextchar\new@ifnextchar
  \array{#1}}
\makeatother

\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=0in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in

\definecolor{faded}{rgb}{.75,.75,.75}
\tikzstyle{f} = [faded]

% Change this line to \def\bold{\mathbf} for ``upright boldface.''
\def\bold{\bm}

\newcommand{\BBB}{\mathcal{B}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\def\0{\bold{0}}
\def\A{\bold{A}}
\def\B{\bold{B}}
\def\D{\bold{D}}
\def\F{\bold{F}}
\def\I{\bold{I}}
\def\J{\bold{J}}
\def\M{\mathcal{M}}
\def\N{\bold{N}}
\def\P{\bold{P}}
\def\U{\bold{U}}
\def\0{\bold{0}}
\def\a{\bold{a}}
\def\b{\bold{b}}
\def\c{\bold{c}}
\def\e{\bold{e}}
\def\p{\bold{p}}
\def\q{\bold{q}}
\def\s{\bold{s}}
\def\t{\bold{t}}
\def\u{\bold{u}}
\def\v{\bold{v}}
\def\w{\bold{w}}
\def\x{\bold{x}}
\def\<{\langle}
\def\>{\rangle}
\def\hat{\widehat}
\newcommand\longto{\longrightarrow}

\def\Sum{\sum\limits}
\def\Prod{\prod\limits}
\def\Cap{\mathop\cap\limits}
\def\Cup{\mathop\cup\limits}
\def\Times{\mathop\times\limits}
\def\Oplus{\mathop\oplus\limits}
\def\Lim{\mathop\lim\limits}

\def\And{\wedge}
\def\Or{\vee}
\newcommand{\Not}[1]{\overline{#1}}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\Span}{Span}
\DeclareMathOperator{\rank}{rank}
\newcommand{\vv}[2]{\begin{bmatrix} #1 \\ #2 \end{bmatrix}}
\newcommand{\vvv}[3]{\begin{bmatrix} #1 \\ #2 \\ #3 \end{bmatrix}}

\newcommand{\ds}{\displaystyle}

\renewcommand{\footrulewidth}{1pt}
\newenvironment{proof}[1][Proof.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{sol}[1][Solution.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{fminipage}
{\setlength{\fboxsep}{15pt}\begin{Sbox}\begin{minipage}}
{\end{minipage}\end{Sbox}\fbox{\TheSbox}}
\newcommand{\qed}{\hfill \ensuremath{\Box}}
\newcommand{\disp}{\displaystyle}

\setlength{\headheight}{15pt}
\lhead{{\sf Bridge course (Algebra)}: Wednesday July 24}
\chead{}\rhead{\thepage}
\lfoot{Due Thursday, July 25 2024}
\cfoot{}\rfoot{Written by M.~Macauley}
  \pagestyle{fancy}

\begin{document}

\noindent \textbf{Topic}:  Jordan canonical form. \\

\noindent \textbf{Do}: Answer the following questions.  Assume that all matrices are over the field $K=\mathbb{C}$. \\

\begin{enumerate}

%%------------------------------------------------------------------------

\item Consider the following Jordan blocks with eigenvalue $\lambda$:
  \[
  \J_2=\begin{bmatrix}\lambda&1\\0&\lambda\end{bmatrix},\qquad
  \J_3=\begin{bmatrix}\lambda&1&0\\0&\lambda&1\\0&0&\lambda\end{bmatrix},\qquad
  \J_4=\begin{bmatrix}\lambda&1&0&0\\0&\lambda&1&0\\0&0&\lambda&1\\
  0&0&0&\lambda\end{bmatrix}.
  \]
  For each $\J_i$, compute $\J_i^2$ and $\J_i^3$. Guess the form of
  $\J_i^k$. Set $k=0$ to find $\J_i^0$ and $k=-1$ to find $\J_i^{-1}$. \\

%%------------------------------------------------------------------------

\item Let $\A$ be a $7\times 7$ matrix over $\C$ with minimal
  polynomial $m(t)=(t-1)^3(t-2)^2$.
\begin{enumerate}
\item List all possible Jordan canonical forms of $A$ up to similarity.
\item For each matrix from Part~(a), find the rank of $(\A-\I)^k$ and 
$(\A-2\I)^k$, for $k\in\mathbb{N}$. \\
\end{enumerate}

%%------------------------------------------------------------------------

\item Let $A$ be an $n\times n$ matrix over $\C$. The matrix $\A$ is
  \emph{nilpotent} if $\A^k=0$ for some $k\in\mathbb{N}$, and $\A$ is \emph{idempotent} if $\A^2=\A$.
  \begin{enumerate}
    \item Prove that if $\A^k=\A$ for some integer $k>1$, then $\A$ is
    diagonalizable. 
  \item Prove that idempotent matrices are similar if and only if they
    have the same trace.
  \item Prove that if $\A$ is nilpotent, then $A^n=0$. 
  \item Prove that if $\A$ is nilpotent, then there is some $r\in\mathbb{N}$
    and positive integers $k_1\geq\cdots\geq k_r$ with
    $k_1+\cdots+k_r=n$ that determine $A$ up to similarity.
  \item Suppose $\A$ and $\B$ are $6\times 6$ nilpotent matrices with the
  same minimal polynomial and $\dim N_{\A}=\dim N_{\B}$. Prove that $\A$ and
  $\B$ are similar. Show by example that this can fail for $7\times 7$
  matrices. 
  \end{enumerate}

%%------------------------------------------------------------------------ 


%%------------------------------------------------------------------------ 

\end{enumerate}

\end{document}
