\documentclass{amsart}


\title{Matrix theory autumn 2011: midterm (take home part)}

\author{James Cummings}


\begin{document}

\maketitle


   This exam is due by noon on Friday 21 October. It should either be delivered to my office (Wean Hall 7101)
   or by email (PDF and JPEG are the only acceptable formats).  Your answers should be typeset or written 
   in black on alternate lines.  You may ask me for clarification of a question but may not consult anyone
   else. You may consult any written or online source, but you must cite any source other than course  
   notes or previous HW sets and solutions. All questions carry equal weight. 
   
   All matrices and vectors are assumed to have real entries.  

\begin{enumerate}

\item  Let $n > 0$. Prove that for any $n \times n$ matrix $A$, the following statements are equivalent:
\begin{enumerate}
\item   $A B = B A$ for all $n \times n$ matrices $B$.
\item   $A B = B A$ for all invertible  $n \times n$ matrices $B$.
\end{enumerate} 

Hint:  You may find it helpful to identify the class of matrices $A$ which satisfy the first property. 



\item  Let $n > 0$ and let $A$ be an $n \times n$ matrix. For all $t \ge 0$, let $N_t$ be the nullspace of $A^t$,
     where by convention $A^0 = 1_{n \times n}$. 

   Prove that:
\begin{enumerate}

\item  $N_t \subseteq N_{t+1}$ for all $t$.

\item  The dimension of $N_t$ (the nullity of $A^t$) is eventually constant, that is there is a number 
       $d$ such that $dim(N_t) = d$ for all sufficiently large $t$.

\item  If $T$ is the  least $t$ such that $dim(N_t) = d$, then  $T \le  d$.
  
\end{enumerate}

   Prove that for all integers $n > 0$, $d$ and $T$ such that $0 < T \le d \le n$
   there is a matrix $A$ such that the nullity of $A^t$ is $d$ for all $t \ge T$,
   and $T$ is the least $t$ such that the nullity of $A^t$ is $d$. 

   
\item  Let $0 < m < n$.  Let $A$ be $n \times m$ and let $B$ be $m \times n$. Prove that
       $A B$ is not invertible. Is it true in general that $B A$ is not invertible? 



\item    For the purposes of the following question we will identify the real
    number $r$ with the $1 \times 1$ matrix whose only entry is $r$. In particular
    when we write ``$M \ge 0$'' when $M$ is $1 \times 1$ we mean that the entry of
    $M$ is non-negative.

    Let $m > 1$ and let $X$ be a $1 \times m$ matrix, that is a row vector of length $m$. 
    Prove that:   

 \begin{enumerate}

\item  $X X^T \ge 0$.

\item  $X X^T$ has rank one if $X \neq 0$. 

\item  $X^T X$ is symmetric.

\item  $A (X^T X) A^T \ge 0$ for all row vectors $A$ of length $m$.

\end{enumerate} 

\end{enumerate} 

\end{document}  


