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\author{Math 25}

\title{Homework 11 --- due Dec 15th}

\begin{document}

\maketitle
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There are two sections: A and B.
Please return each part
to the proper CA.

\renewcommand{\thesection}{\Alph{section}}

\section{Problems graded by Benjamin}

In the following 4 problems, $K$ is a field which contains $\QQ$ and
$M_n(K)$ is the set of all $n \times n$ matrices.

\subsection{Matrices of trace $0$ I} Let $M \in M_n(K)$ be a matrix whose trace is
$0$. 
\begin{enumerate}
\item
Prove that there exists an invertible matrix $P$ such that the
first column of $P^{-1}MP$ is \[ \begin{pmatrix} 0 \\ 1 \\ 0 \\ \vdots
  \\ 0 \end{pmatrix}. \]
\item 
Prove that there exists an invertible matrix $Q$ such that all the
diagonal entries of $Q^{-1}MQ$ are $0$.
\end{enumerate}

\subsection{Matrices of trace $0$ II} Let \[ D =  \begin{pmatrix} 1 & &  & \\ & 2 & &  \\ & &
  \ddots & \\ & & & n \end{pmatrix} \] and let $\phi:M_n(K) \ra
  M_n(K)$ be the map defined by $\phi(M)=MD-DM$.
\begin{enumerate}
\item Prove that $\ker \phi$ is the set of diagonal matrices. What is
  the rank of $\phi$?
\item  Prove that $\on{im} \phi$ is a subspace of the space of 
  matrices whose diagonal entries are $0$.
\item Prove that $\on{im} \phi$ is the space of 
  matrices whose diagonal entries are $0$.
\end{enumerate}

\subsection{Matrices of trace $0$ III (Shoda's theorem)} 
Using the previous two problems, prove that a matrix $M$
  satisfies $\on{Tr}(M)=0$ if and only if there exist $A, B \in
  M_n(K)$ such that $M=AB-BA$.

\subsection{Matrices of trace $0$ IV} Prove that the subspace of $M_n(K)$ generated by all
  nilpotent matrices is the space of matrices whose trace is $0$.

\subsection{Skolem-Noether's theorem} 
Let $\phi : M_n(K) \ra M_n(K)$ be a bijective map such that
$\phi(A+\lambda B)=\phi(A)+\lambda \phi(B)$, $\phi(\on{Id})=\on{Id}$ and
$\phi(AB)=\phi(A)\phi(B)$. Such a map is called an automorphism of
$M_n(K)$. The goal of the problem is to find all automorphisms of
$M_n(K)$. 

\begin{enumerate}
\item Check that if $P$ is an invertible matrix, then the map $\phi_P$
defined by $\phi_P(M)=PMP^{-1}$ is an automorphism of
$M_n(K)$.
\item 
Let $E_{i,j}$ be the matrix all of whose entries are $0$ except
the $(i,j)$-th which is $1$. Compute $E_{i,j}E_{k,\ell}$ (the answer
depends on whether $j=k$ or not).
\item Now let $\phi$ be an unspecified automorphism and
let $U_{i,j}=\phi(E_{i,j})$. Prove that $\ker
(U_{1,1}-\on{Id})$ is non-trivial.
\item Let $f_1 \in \ker (U_{1,1}-\on{Id})$ be a non-trivial element
  and for $k \geq 2$, set $f_k=U_{k,1}(f_1)$. Prove that
  $\{f_1,\cdots,f_n\}$ is a basis of $K^n$. 
\item Let $P$ be the matrix of $\{f_i\}$ in the usual basis of
  $K^n$. Prove that $\phi$ and $\phi_P$ coincide on the
  $E_{k,1}$'s. 
\item prove by recurrence on $n$ that if $\phi$ 
is an automorphism of
$M_n(K)$, then there exists an invertible matrix $Q$ such that
$\phi(M)=QMQ^{-1}$. 
\end{enumerate}

\section{Problems graded by Inna}

\subsection{Symmetries} 
Let $f \in \on{End}(V)$ be a map such that $f^2=\on{Id}$. 
\begin{enumerate}
\item Prove that $f$ is diagonalizable, with eigenvalues $\pm 1$. Let
  $V_{\pm}$ be the eigenspace corresponding to $\pm 1$.
\item Why is such a map called a symmetry (your answer should involve
  $V_{\pm}$)?
\end{enumerate}

\subsection{} Prove that if $f \in \on{End}(V)$, then 
$V = \on{ker}(f) \oplus \on{im}(f)$ if and only if $0$ is a simple
root of $\Pi_f$.

\subsection{Projections} Let $f \in \on{End}(V)$ be a map such that
$f^2=f$. 
\begin{enumerate}
\item Prove that $\on{ker}(f) \oplus \on{im}(f)  = V$.
\item Prove that $f$ is diagonalizable. What are its eigenvalues, and
  what are the corresponding eigenspaces?
\item Why is such a map called a projection?
\end{enumerate}

\subsection{} Let $\CC_n[X]$ be the space of polynomials with complex
coefficients and degree $\leq n$. Prove that the map $f: \CC_n[X] \ra
\CC_n[X]$ defined by $f(P)=(X^2-1)P^{''}$ is diagonalizable.

\subsection{} Let $A$ and $B$ be two matrices and let \[ M =
\begin{pmatrix} A & \\ & B \end{pmatrix}. \] Find $\Pi_M$ in terms of
$\Pi_A$ and $\Pi_B$.

\subsection{} Prove that if $f:E \ra E$ is a diagonalizable map and if
$V \subset E$ is a subspace stable by $f$, then the restriction of $f$
to $V$ is diagonalizable. Prove an analogous statement for
``triangulable''. 

\subsection{}
Let $M,N \in M_n(K)$ be such that $MN-NM=\lambda M$ where $\lambda
\neq 0$. 
\begin{enumerate}
\item Prove that $M^kN-NM^k = k \lambda M^k$ for all $k \geq 0$.
\item Prove that $M$ is nilpotent.
\end{enumerate}


\end{document}

