% LaTeX file -- M245, Fall 2012, Assignment 3

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\begin{document}
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%%%%%%%   TITLE   %%%%%%
\begin{center}
\textbf{\large Math 245 \\ Assignment 3\\
Due Friday October 19}
\end{center}
\vskip 2em 

\begin{Ex}

\item \begin{parts}
\item Let $A,B \in \L(V)$. Prove that $\nul(AB) \le \nul(A) + \nul(B)$.
\item Prove that $T$ is diagonalizable if and only if the minimal polynomial $m_T$ splits into
linear factors and has no repeated roots.
\end{parts}


\item Find the possible Jordan forms that $T$ could have given
the following information. \vspace{-3ex}
\begin{parts}
\item  $p_T(x)=(x-4)^4(x+1)^2$ and $m_T(x)=(x-4)^3(x+1)^2$.
\item  $p_T(x)=(x-4)^4(x+1)^2$ and $m_T(x)=(x-4)^2(x+1)$.
\end{parts}



\item \begin{parts}
\item Find the Jordan form for 
\[
 T=
 \left[\begin{array}{rrrrr}
 1&0&3&0&5\\
 0&2&0&1&0\\
 3&0&9&0&15\\
 0&-1&0&0&0\\
 -2&0&-6&0&-10
 \end{array}\right].
\]
\item Find an explicit $S$ so that $STS^{-1}$ is in Jordan form.
\item Hence find a square root for $T$.
\end{parts}


\item Let $T = J(0,2) \oplus J(2,3) \oplus J(3,1)$. Find a polynomial $p$ so that
$p(T) = A$ where
\[
 T = 
\left[\begin{array}{rr|rrr|r}
 0&0&0&0&0&0\\
 1&0&0&0&0&0\\ \hline
 0&0&2&0&0&0\\
 0&0&1&2&0&0\\
 0&0&0&1&2&0\\ \hline
 0&0&0&0&0&3
\end{array}\right]
\qand
 A =
\left[\begin{array}{rr|rrr|r}
 0&0&0&0&0&0\\
 0&0&0&0&0&0\\ \hline
 0&0&1&0&0&0\\
 0&0&0&1&0&0\\
 0&0&1&0&1&0\\ \hline
 0&0&0&0&0&0
\end{array}\right]
\]


\item Suppose that $x$ is a \textit{cyclic vector} for $T$, meaning that $\{x,Tx,T^2x,\dots\}$ span $V$.
\begin{parts}
\item Show that $\{x,Tx,\dots,T^{n-1}x\}$ is a basis.
\item Prove that  $m_T=p_T$.
\end{parts}


\item \textbf{Bonus}. Let $T\in\L(V)$ over an arbitrary field $\bF$.
\begin{parts}
\item  Show that $T$ has a block upper triangular form so that the diagonal entries $T_i$, $1 \le i \le k$,
each have a cyclic vector.
\item Let $p_i(x)=p_{T_i}(x)$. Show that each $p_i$ divides $m_T$.
Hence deduce that every irreducible factor of $p_T$ is a factor of $m_T$.
\end{parts}

\end{Ex}
\end{document}