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\title{Math 421-621\\Fall 2013\\Homework 6}
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\centerline{\it Due Wednesday, February 27, 2013.}


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\noindent 1. We say that the $n\times n$ matrix $A$ is nilpotent if $A^m=0$ for some positive integer $m$. 
\begin{itemize}
\item[a)] (5 pt) Show that every eigenvalue of a nilpotent matrix is $0$.
\item[b)] (5 pt) Show that if $A$ is nilpotent and $n\times n$, then $A^n=0$.
\end{itemize} 

\centerline{}

\noindent 2. Recall that $A$ and $B$ are similar $n\times n$ if there is a nonsingular $n\times n$ (say $P$) such that $B=P^{-1}AP$.
\begin{itemize}
\item[a)] (5 pt) Show that $\text{det}(A)=\text{det}(B)$.
\item[b)] (5 pt) Show that if $(\lambda, v)$ is an eigenvalue/eigenvector pair for the matrix $A$ then $\lambda$ is an eigenvalue for $B$ (corresponding to what eigenvector of $B$?).  
\end{itemize}

\centerline{}

\noindent 3. (5 pt) Show that the $n\times n$ matrix $A$ is similar to a diagonal matrix if and only if $A$ has $n$ linearly independent eigenvectors.

\centerline{}

\noindent 4. (5 pt) Let $V$ be an $n-$dimensional vector space over $\mathbb{F}$ and $\phi:V\longrightarrow V$ a fixed linear transformation. Show that $V$ is a $\mathbb{F}[x]-$module with action defined by

\[
f(x)\cdot v=f(\phi)(v)
\]

\noindent where $f(x)\in\mathbb{F}[x]$ and $v\in V$. 



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