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\title{Math 421-621\\Spring 2013\\Homework 7}
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\centerline{\it Due Wednesday April 3, 2013.}

\centerline{}

\noindent 1. Find the canonical forms (the rational canonical form, primary rational canonical form and Jordan canonical form if possible) for the following matrices over $\mathbb{Q}$:
\begin{itemize}
\item[a)] (15 pt)
$\left[
\begin {array}{cccc}
-1&\ \ 2&-1&\ \ 0\\

-2&\ \ 3&-1&\ \ 0\\

\ \ 1&-1&\ \ 2&\ \ 0\\

-1&\ \ 1&\ \ 0&\ \ 1
\end {array}
\right]$

\item[b)] (15 pt) 
$\left[
\begin {array}{ccccc}
\ \ 3&\ \ 1&\ \ 0&\ \ 1&\ \ 1\\

\ \ 0&\ \ 3&\ \ 0&-1&\ \ 0\\

\ \ 0&-2&\ \ 4&\ \ 2&\ \ 0\\

\ \ 0&-1&\ \ 0&\ \ 3&\ \ 0\\

\ \ 1&-1&\ \ 0&-1&\ \ 3
\end {array}
\right]$

\end{itemize}

\centerline{}



\noindent 3. A matrix $A$ is said to be nilpotent if there is an $m\geq 1$ such that $A^m=0$. Additionally, we define the trace of $A$ ($\text{tr}(A)$) to be the sum of the diagonal elements of $A$. For this problem, you may assume that $A$ is an $n\times n$ matrix over a field $\mathbb{F}$.
\begin{itemize}
\item[a)] (5 pt) Show that $\text{tr}(AB)=\text{tr}(BA)$.
\item[b)] (5 pt) Show that if $P$ is an invertible $n\times n$ matrix then $\text{tr}(P^{-1}AP)=\text{tr}(A)$.
\item[c)] (5 pt) Show that $A$ is nilpotent if and only if all of its eigenvalues are $0$.
\item[d)] (5 pt) Show that if $A$ is nilpotent, then $\text{tr}(A)=0$. 
\item[e)] (5 pt) Determine the status of the converse of the statement in part d).
\end{itemize}




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