\documentclass{amsart}
\title{Problem Set \#3}
\author{Due Date: October 22}

\begin{document}
\maketitle

1. Let $A$ be an $n\times n$-matrix, and let $B$ be an $m\times m$-matrix. Let
\[
C=\begin{bmatrix}
A & 0 \\
0 & B
\end{bmatrix}\in M_{(m+n,m+n)}.
\]
For this problem, you have to show that $|C|=|A|\cdot |B|$, using the definition of the determinant.
\begin{enumerate}
\item Let $\Sigma_n\times\Sigma_m$ be 
\[
\Sigma_n\times\Sigma_m=\big\{f\in\Sigma_{n+m} | f(\{1,\dots,n\})=\{1,\dots,n\}\big\}.
\]
In other words, we write the set from $1$ to $n+m$ as the set from $1$ to $n$ then the set from $n+1$ to $n+m$, and we require that $f$ preserves holds these two subsets invariant. Show that if $\sigma\in \Sigma_{n+m}-\Sigma_n\times\Sigma_m$, then
\[
c_{1,\sigma(1)}\dots c_{(n+m),\sigma(n+m)}=0,
\]
where $c_{i,j}$ is the element in position $i,j$ of $C$. Conclude that
\[
|C|=\sum_{\sigma\in\Sigma_n\times\Sigma_m} {sign(\sigma)}c_{1,\sigma(1)}\dots c_{(n+m),\sigma(n+m)}.
\]
\item Let $\sigma\in\Sigma_n\times\Sigma_m$, let $\sigma_1$ be the restriction of $\sigma$ to $\{1,\dots,n\}$, and let $\sigma_2$ be the restriction of $\sigma$ to $\{n+1,\dots,n+m\}$. Show that
\[
sign(\sigma)=sign(\sigma_1)\cdot sign(\sigma_2).
\]
\item Combine parts 1 and 2 to conclude that
\[
|C|=|A|\cdot |B|.
\]
\end{enumerate}

2. Let $V$ be a 7 dimensional vector space. Show that there is more than one possible Jordan form for a nilpotent operator on $V$ with index of nilpotency $3$ and with a $3$-dimensional kernel.

3. Let $p(\lambda)=\lambda^n+a_{n-1}\lambda^{n-1}+\dots+a_0$. Let 
\[
A_p=\begin{bmatrix}
0 & 0 & \dots & 0 & -a_0 \\
1 & 0 & \dots & 0 & -a_1 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \dots & 0 & -a_{n-2} \\
0 & 0 & \dots & 1 & -a_{n-1}
\end{bmatrix}.
\]

Show that $p_{A_p}(\lambda)=(-1)^n p(\lambda)$. (Hint: Expand the determinant along the first row or column and use induction on $n$).

4. Now some examples to work through. Find the Jordan form $J$ of the following matrices $A$. Find also a Jordan basis and the change of basis matrix $P$ such that $J=P^{-1}AP$.
\begin{enumerate}
\item \[A=\begin{bmatrix}
2 & 1 & 0 & 0 \\
-1 & 4 &0 & 0 \\
-2 & 4 & 0 & 1 \\
-4 & 10 & -9 & 6
\end{bmatrix}.\]
\item \[B=\begin{bmatrix}
2 & 1 & 0 & 0 \\
-1 & 4 & 0 & 0\\
-1 & 1 & 3 & 0\\
-1 & 1 & 0 & 3
\end{bmatrix}.\]
\item \[C=\begin{bmatrix}
2 & 1 & 0 & 0\\
-1 & 4 & 0 & 0\\
-3 & 6 & -1 & 1\\
-6 & 13 & -9 & 5
\end{bmatrix}.\]
\item 
\[
D=\begin{bmatrix}
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
1 & -3 & 3 & 0 & 0 & 0 \\
3 & -9 & 9 & -3 & 1 & 0 \\
7 & -23 & 28 & -16 & 5 & 0 \\
14 & -49 & 65 & -40 & 10 & 1
\end{bmatrix}.
\]

\end{enumerate}
\end{document}