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\title{\vspace{-50pt}Math 210A: Modern Algebra\\
\large Thomas Church ({\tt tfchurch@stanford.edu})\\
\href{http://math.stanford.edu/~church/teaching/210A-F17/}{\nolinkurl{http://math.stanford.edu/~church/teaching/210A-F17}}}
\author{}
\date{}

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\begin{document}
\maketitle
\vspace{-50pt}
\begin{center}
{\huge \bf Homework 6}\\\vspace{8pt}
{\Large Due Thursday night, November 2} {\footnotesize(technically 5am Nov.\ 3)}\\
\end{center}

\begin{question} Let $A$ and $B$ be two $n\times n$ matrices with entries in a field $K$.\newline
Let $L$ be a field extension of $K$, and suppose there exists $C\in \GL_n(L)$ such that $B=CAC^{-1}$.\newline Prove there exists $D\in \GL_n(K)$ such that $B=DAD^{-1}$.
\newline
{\footnotesize (that is, turn the argument sketched in class into an actual proof)}
\end{question}

\begin{question}
Let $k$ be an algebraically closed field of characteristic $\neq 2$. Fix two nonzero elements $\lambda,\mu\in k$. Let $V$ and $W$ be 2-dimensional $k$-vector spaces. Let $\alpha\colon V\to V$ have matrix $\begin{psmallmatrix}\lambda&0\\1&\lambda\end{psmallmatrix}$, and let $\beta\colon W\to W$ have matrix $\begin{psmallmatrix}\mu&0\\1&\mu\end{psmallmatrix}$. Let $\gamma\colon V\otimes W\to V\otimes W$ be $\alpha\otimes \beta$ (defined on elementary tensors by $\gamma(v\otimes w)=\alpha(v)\otimes \beta(w)$).

Find the Jordan decomposition of $\gamma$ (that is, give a list of blocks and their sizes, and prove your answer is correct). Give a basis for all eigenspaces of $\gamma$. What happens if $\text{char }k = 2$?
\end{question}


\begin{question}
Let $k$ be an algebraically closed field of characteristic $\neq 3$. Let $V$ be an $n$-dimensional $k$-vector space, and suppose that $T\colon V\to V$ has minimal polynomial $(t-\lambda)^n$ for some nonzero $\lambda\in k$. Find the Jordan decomposition of $T^3$.
\end{question}

\begin{question}
Let $V$ be a finite-dimensional nonzero vector space over a field $k$.  

\begin{enumerate}
\item[(a)] For each monic irreducible $\pi \in k[t]$, define \[V(\pi)=\big\{\ v\in V\ \big\vert\ \exists k\in \N \text{ s.t. }(\pi(T))^k(v)=0\ \big\}.\]
\noindent{\small  (When $k$ is algebraically closed, these are the {\em generalized eigenspaces} $V_\lambda=V(t-\lambda)$ of $T$.)}

Prove that $V(\pi) \ne 0$ if and only if $\pi|m_T$, and that $V = \bigoplus_{\pi|m_T} V(\pi)$.
\end{enumerate}
An endomorphism $T\colon V \rightarrow V$ is \emph{semisimple} if
every $T$-stable subspace of $V$ admits a $T$-stable complementary subspace: i.e.\ for every $T(U) \subseteq U$
 there exists a decomposition $V = U \oplus W$ with $T(W) \subseteq W$.\newline {\small (Keep in mind that such a complement
is not unique in general; e.g.\ consider scalar multiplication by 2.)}
\begin{enumerate}
\item[(b)] Use rational canonical form to
prove that $T$ is semisimple if and only if $m_T$ has no repeated irreducible factor over $k$. 
{\small (Hint: apply (a) to $T$-stable subspaces of $V$ to reduce to the case when $m_T$ has one monic irreducible factor.)}
Deduce that\begin{enumerate}
\item[(i)] a Jordan block of rank $> 1$ is never semisimple,
\item[(ii)] if $T$ is semisimple then $m_T$ is the ``squarefree part'' of
$\chi_T$, and 
\item[(iii)] if  $T$ is semisimple and $U \subseteq V$ is a $T$-stable nonzero proper subspace
then the induced endomorphisms $T_U\colon U \rightarrow U$ and
$\overline{T}\colon V/U \rightarrow V/U$ are semisimple. 
\end{enumerate}
\item[(c)] Let $V'$ be another nonzero  finite-dimensional $k$-vector space, and let $T'\colon V' \rightarrow V'$ be another endomorphism. Prove that $T$ and $T'$ are semisimple if and only if the endomorphism 
$T \oplus T'$ of $V \oplus V'$ is semisimple. 
\end{enumerate}

\end{question}


\begin{question}
Let $V$ be a $n$-dimensional $k$-vector space with $0<n<\infty$, and let $T\colon V \rightarrow V$ be an endomorphism.
\begin{enumerate}[label=(\alph*)]
\item  Using rational canonical form and Cayley--Hamilton,
prove the following are equivalent:
\begin{enumerate}
\item $\exists k\geq 1$ such that $T^k = 0$.
\item $T^n = 0$.
\item There is an ordered basis of $V$ w.r.t.\ which the matrix for $T$ is upper triangular with 0's on the diagonal.
\item $\chi_T = t^n$.
\end{enumerate}
We call such $T$ {\em nilpotent}.

\item We say that $T$ is {\em unipotent} if $T - 1$ is nilpotent.  Formulate characterizations of unipotence analogous to the conditions in (a),
and prove that a unipotent $T$ is invertible. 

\item Assume $k$ is algebraically closed.  Using Jordan canonical form and generalized eigenspaces, prove that 
there is a unique expression
\[T = T_{\rm{ss}} + T_{\rm{n}}\] where $T_{\rm{ss}}$ and $T_{\rm{n}}$ are a pair of {\em commuting}
endomorphisms of $V$ with $T_{\rm{ss}}$ semisimple and $T_{\rm{n}}$ nilpotent.  (This is the {\em additive
Jordan decomposition} of $T$.)

Show in general that
$\chi_T = \chi_{T_{\rm{ss}}}$ (so $T$ is invertible if and only if $T_{\rm{ss}}$ is invertible)

Show by example with $\dim V = 2$ that uniqueness fails if we drop
the ``commuting'' requirement. {\footnotesize (You just need to give the matrix $T$ and the two decompositions $T=T_{\rm{ss}} + T_{\rm{n}}$ and $T=T'_{\rm{ss}} + T'_{\rm{n}}$; you do not need to prove these matrices are semisimple/nilpotent, as long as they are.)}

\item Assume $k$ is algebraically closed and $S\colon V\to V$ is invertible.  Using the existence and uniqueness of additive Jordan decomposition,
prove that there is a unique expression
\[S = S_{\rm{ss}} S_{\rm{u}}\] where $S_{\rm{ss}}$ and $S_{\rm{u}}$ are  {\em commuting}
endomorphisms of $V$ with $S_{\rm{ss}}$ semisimple and $S_{\rm{u}}$ unipotent (so $S_{\rm{ss}}$ is necessarily
invertible too).  This is the {\em multiplicative Jordan decomposition}~of~$T$. 
\end{enumerate}
\end{question}



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