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\author{Math 25}

\title{Final exam}

\begin{document}

\maketitle

\section*{Instructions: read carefully}

This exam starts at 9:15 and finishes at 12:15.

\begin{itemize}
\item You \textbf{are} allowed to use: your notes from class, 
anything that's on the course's website or on the CA's website. It is
your responsibility to check that what you're using is indeed
mathematically correct.
\item
You \textbf{are not} allowed to use: (text)books, photocopies,
a calculator or a computer. 
\end{itemize}

You can use scrap paper but I
will only grade what's in the ``blue books''. 

\vspace{40pt}

Name:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

Return this sheet with your solutions. 

\vspace{20pt}
\begin{tabular}{||c||c|c|c||c|c||c|c|c|c|c||c||}
\hline
{\S} & 1.1 & 1.2 & 1.3 & 2.1 & 2.2 & 3.1 & 3.2 & 3.3 & 3.4 & 3.5 & Total \\
\hline 
Points & & & & & & & & & & & \\
\hline 
Out of & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 100 \\
\hline
\end{tabular}
\vspace{20pt}

\section{Simultaneous Diagonalisation}\label{codiag}
Let $f$ and $g$ be two endomorphisms of 
a $d$-dimensional vector space $V$ over a field $K$. We say
that $f$ and $g$ are \emph{simultaneously diagonalisable} if there
exists a basis $e_1,\cdots,e_d$ of $V$ and elements
$\{\lambda_i\}_{i=1 \cdots d}$ and $\{\mu_i\}_{i=1 \cdots d}$ of $K$
such that $f(e_i)=\lambda_i e_i$ and $g(e_i)=\mu_i e_i$ for $i=1
\cdots d$. 

Observe that if $f$ and $g$ are simultaneously
diagonalisable, then $fg=gf$.
In the rest of this problem, we will assume that $f$ and $g$ are
diagonalisable and that $fg=gf$. 
The goal of the problem is to prove that this implies that 
$f$ and $g$ are simultaneously diagonalisable.

\subsection{10 pts} Prove that if $W$ is a sub-vector space of $V$,
stable under $f$, then the restriction of $f$ to $W$ is
diagonalisable.  

\subsection{10 pts} Prove that if $\mu \in K$, then the vector
space $\on{ker}(g-\mu \on{Id})$ is stable under $f$.

\subsection{10 pts} Prove that $f$ and $g$ are simultaneously
diagonalisable. 

\section{Topology of some spaces of matrices}

In the remainder of this exam
we will study the topology of certain
spaces of matrices with complex coefficients. The absolute value
$|\cdot|$ is the usual one on $\CC$. We will need to use
three topological spaces:
\begin{itemize}
\item The space $\CC^n$ is the space of $n$-uples
$(x_1,\cdots,x_n)$ with the distance defined by $d(x,y)= \sup_{1 \leq i
  \leq n} |x_i-y_i|$ if $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,y_n)$.

\item The space $\CC[X]_n$ is the set of polynomials $P(X)=X^n + p_{n-1} X^{n-1} + 
\cdots + p_0$ with the distance defined by $d(P,Q)= \sup_{0 \leq i
  \leq n-1} |p_i-q_i|$ if 
$P(X)=X^n + p_{n-1} X^{n-1} +  \cdots + p_0$ and $Q(X)=X^n + q_{n-1} 
X^{n-1} +  \cdots + q_0$.

\item The space $\on{M}(n,\CC)$ is the space of all $n \times n$ 
matrices with complex coefficients. If $A=(a_{ij})_{1 \leq i,j \leq
n}$ and $B=(b_{ij})_{1 \leq i,j \leq
n}$, we define $d(A,B) = \sup_{1 \leq i,j \leq n} |a_{ij}-b_{ij}|$.
\end{itemize}

\subsection{10 pts}\label{cont}
Explain \emph{briefly} 
why the following maps are continuous. You should not use
$\epsilon/\delta$ arguments; no more than a few lines of justification
are necessary:
\begin{enumerate}
\item $\det: \on{M}(n,\CC) \ra \CC$.
\item the ``characteristic polynomial'' map $\chi : \on{M}(n,\CC) \ra
  \CC[X]_n$ which maps $M$ to $\chi_M = \on{det}(X \on{Id} - M)$. 
\item the map $f : \CC^n \ra \CC[X]_n$ which maps $(z_1,\cdots,z_n)$
  to $\prod_{i=1}^n (X-z_i)$.
\item the ``conjugation by $P$'' map $M \mapsto PMP^{-1}$ from
  $\on{M}(n,\CC)$ to itself, where $P$ is an invertible matrix.
\end{enumerate}

\subsection{10 pts}\label{rk1} Prove that the set of invertible 
matrices $M \in \on{M}(n,\CC)$ is open in $\on{M}(n,\CC)$. 

\section{Diagonalisable matrices}

Let $\on{D}(n,\CC)$ be the subset 
of $\on{M}(n,\CC)$ consisting
of diagonalisable matrices. We will show that $\on{D}(n,\CC)$ is dense
in $\on{M}(n,\CC)$ and we will compute its interior.

\subsection{10 pts} Let $M$ be a matrix which is upper
triangular. Prove that each diagonal entry is an eigenvalue of $M$.

\subsection{10 pts} Prove that $\on{D}(n,\CC)$ is
dense in $\on{M}(n,\CC)$.

\subsection{10 pts} Prove that if $E$ is a non-empty subset of
$\CC^n$ then $E$ is bounded if and only if $f(E) \subset \CC[X]_n$ 
is bounded (here $f$ is the map from (3) of exercise \ref{cont}).

\subsection{10 pts} Use this to prove that the set of polynomials 
having only simple roots is open in $\CC[X]_n$. 

\subsection{10 pts} Prove that
the interior of $\on{D}(n,\CC)$ in 
$\on{M}(n,\CC)$ is the set of matrices having
$n$ distinct eigenvalues. 
\end{document}