\documentclass{amsart}


\title{Matrix theory autumn 2011: homework 8}

\author{James Cummings}


\begin{document}

\maketitle
    
   Homework due at start of class on {\em Friday} 18 November. 


   Recall that if $x, y \in {\mathbb C}^n$ then 

\[
     x. y = \sum_{i=1}^n x_i \overline{y_i}
\]
 and
\[
    \Vert x \Vert^2 = x . x
\]

   Some of these questions are about real linear algebra and some are about complex linear algebra. To avoid
   any confusion I have marked each question as ``real'' or ``complex''. 


\begin{enumerate}


\item  (Complex) Exactly as for ${\mathbb R}^n$, a set $X$ of vectors in ${\mathbb C}^n$ is {\em orthonormal}
       if for all $x, y \in X$ we have $x. y = 1$ for $x = y$ and $x. y = 0$ for $x \neq y$. 
        
       Prove that if $v_1, \ldots v_n$ enumerates an ON set in ${\mathbb C}^n$ then for every
       $v \in {\mathbb C}^n$ we have $v = \sum_i (v. v_i) v_i$. What would be wrong with the formula
       ``$v = \sum_i (v_i. v) v_i$''? 


\item (Complex)     Find an ON basis for the subset of ${\mathbb C}^4$ spanned by the vectors
       $(1, i, 0, 1)$, $(0, 2, -i, 3)$ and $(i, 1, -1, 2)$. 


\item  (Real) Let $T: {\mathbb R}^2 \rightarrow {\mathbb R}^2$ be an orthogonal linear map.
       Prove that $T$ is either a rotation about the origin or a reflection in a line through the
       origin. 

       Hint:  $T$ is determned by $T e_1$ and $T e_2$. 

 
\item  (Real) Prove that if $n$ is odd then every $n \times n$ real matrix has at least one 
       eigenvector. 

       Hint: The characteristic polynomial has odd degree. Ask Paul or consult the internet
       if you don't see why this is helpful.

\item  (Real) Let $A$ be an orthogonal $n \times n$ matrix. Prove that every eigenvalue of $A$
       is either $1$ or $-1$, and that $det(A)$ is $1$ or $-1$. 


\item  (Real)  Let $v \in {\mathbb R}^3$ be a nonzero vector.  
 Describe (with proof) in geometrical terms the orthogonal linear maps 
        $T:{\mathbb R}^3 \rightarrow {\mathbb R}^3$
   such that $Tv = v$.

 Hint: knowing what the orthogonal maps from ${\mathbb R}^2$ to ${\mathbb R}^2$ are
   like may be useful.  


\item   (Real)  Describe (with proof) in geometrical terms all  the orthogonal linear maps 
        $T:{\mathbb R}^3 \rightarrow {\mathbb R}^3$.

  Hint: Start by combining some of the previous exercises to prove that if
    $R$ is the orthogonal map $R(x, y, z) = (-x, -y, -z)$, then one 
    of the orthogonal maps $T$ and $R \circ T$ fixes at least one nonzero
    vector $v$. 


\end{enumerate} 




\end{document}



