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\pagestyle{fancy} \lhead{{\sf Homework 8 $|$ Due October 24
    (Tuesday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Fall 2023 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Lax, Appendix 15, pages 363--366. \\

\begin{enumerate}

%%------------------------------------------------------------------------ 

\item Let $A$ be a $7\times 7$ matrix over $\C$ with minimal
  polynomial $m(t)=(t-1)^3(t-2)^2$.
\begin{enumerate}
\item List all possible Jordan canonical forms of $A$ up to similarity.
\item For each matrix from Part~(a), find the rank of $(A-I)^k$ and 
$(A-2I)^k$, for $k\in\N$. \smallskip
\end{enumerate}

%%------------------------------------------------------------------------ 

\item Let $A$ be an $n\times n$ matrix over $\C$. The matrix $A$ is
  \emph{nilpotent} if $A^k=0$ for some $k\in\N$.
  \begin{enumerate}
  \item Prove that if $A$ is nilpotent, then $A^n=0$. 
  \item Prove that if $A$ is nilpotent, then there is some $r\in\N$
    and positive integers $k_1\geq\cdots\geq k_r$ with
    $k_1+\cdots+k_r=n$ that determine $A$ up to similarity.
\item Suppose $A$ and $B$ are $6\times 6$ nilpotent matrices with the
  same minimal polynomial and $\dim N_A=\dim N_B$. Prove that $A$ and
  $B$ are similar. Show by example that this can fail for $7\times 7$
  matrices. \smallskip
  \end{enumerate}

%%------------------------------------------------------------------------ 

\item Let $A$ and $B$ be $n\times n$ matrices over $\C$. The matrix
  $A$ is \emph{idempotent} if $A^2=A$.
\begin{enumerate}
  \item Prove that if $A^k=A$ for some integer $k>1$, then $A$ is
    diagonalizable. 
  \item Prove that idempotent matrices are similar if and only if they
    have the same trace.
  \item Prove that if $A$ and $B$ are idempotent and $B=UAV$ holds for
    some invertible maps $U,V\colon X\to X$, then $A$ and $B$ are
    similar. \smallskip
\end{enumerate}

%%------------------------------------------------------------------------ 

\item Consider the matrices
  $A=\begin{bmatrix}2&0&0\\0&2&0\\0&0&3\end{bmatrix}$ and
  $B=\begin{bmatrix}5&-3&0\\-3&5&0\\0&0&2\end{bmatrix}$.
  \begin{enumerate}
  \item Decompose $\R^3$ into a direct sum of eigenspaces of each matrix. 
  \item Further decompose the $2$-dimensional $A$-eigenspace as a
    direct sum of two $1$-dimensonal $B$-eigenspaces, and vice-versa.
  \item Write $\R^3$ as a direct sum of three $1$-dimensional
    subspaces that are common eigenspaces of $A$ and $B$, two
    different ways.
  \item For each of your answers to Part~(c), find a matrix $P$ so
    that $P^{-1}AP=D_A$ and $P^{-1}BP=D_B$, where $D_A$ and $D_B$ are
    diagonal. \smallskip
  \end{enumerate}
  
%%------------------------------------------------------------------------ 

\item Let $X$ be an $n$-dimensional vector space over $\C$, and let
  $A,B\colon X\to X$ be linear maps.
  \begin{enumerate}
  \item Prove that if $AB=BA$, then for any eigenvector $v$ of
    $A$ with eigenvalue $\lambda$, the vector $Bv$ is an eigenvector
    of $A$ for $\lambda$.
  \item Suppose that $A$ and $B$ are both diagonalizable. Prove
    that $AB=BA$ if and only if they are \emph{simultaneously
      diagonalizable}, i.e., there exists an invertible $n\times
    n$-matrix $P$ such that both $P^{-1}AP$ and $P^{-1}BP$ are
    diagonal matrices. 
  \item Show that if $\{A_1,\dots,A_k\mid A_i\colon X\to X\}$ is a set
    of pairwise commuting maps, then there is a nonzero $x\in X$ that
    is an eigenvector of every $A_i$.
  \end{enumerate}

%%------------------------------------------------------------------------ 

\end{enumerate}

\end{document}







  
