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\pagestyle{fancy} \lhead{{\sf Homework 11 $|$ Due April 18
    (Tuesday) }} \rhead{\thepage} \cfoot{{\sf Math 853 $|$ Linear
    Algebra $|$ Spring 2017 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Lax, Chapter 7, pages 89--100. \\

\begin{enumerate}


  % Lax Exercise 7.15 (p. 96)
\item Let $X$ be the space of continuous complex-valued functions on
  $[-1,1]$ and define an inner product on $X$ by
  \[
  (f,g)=\int_{-1}^1f(s)\overline{g(s)}\,ds\,.
  \]
  Let $m(s)$ be a continuous function of absolute value $1$, that is,
  $|m(s)|=1$, $-1\leq s\leq 1$. Define $M$ to be multiplication by $m$:
  \[
  (Mf)(s)=m(s)f(s)\,.
  \]
  Show that $M$ is unitary. \\

  
  
  % July 2009 B1(a), and July 2008 B2(c)
\item Let $A$ be a linear map of a finite-dimensional complex
  inner product space $X$.
  \begin{enumerate}
  \item A matrix is \emph{normal} if $AA^*=A^*A$. It is unitarily
    similar to a diagonal matrix if $A=U^*DU$ for a diagonal matrix
    $D$ and unitary matrix $U$. Show that these conditions are
    equivalent.
  \item Prove that if $A$ is normal then it has a square-root, that
    is, a matrix $B$ such that $A=B^2$. Is $B$ necessarily normal?
    Unique?
  \item Suppose that $A$ is diagonalizable. Prove that $A$ is normal
    if and only if each eigenvector of $A$ is an eigenvector of $A^*$. \\
  \end{enumerate}


  
%\item Express
%  $q(x_1,x_2,x_3)=3x_1^2+8x_1x_2-7x_1x_3+12x_2^2-8x_2x_3+6x_3^2$ as
%  $q(x)=x^TAx$, where $A$ is symmetric. \\


 % Lax Exercise 8.11 (p. 119)
\item Let $S$ be the cyclic shift mapping of $\C^n$, that is,
  $S(z_1,\dots,z_n)=(z_n,z_1,\dots,z_{n-1})$.
  \begin{enumerate}
  \item Prove that $S$ is unitary.
  \item Determine the eigenvalues and eigenvectors of $S$.
  \item Find an orthonormal basis of $\C^n$ consisting of eigenvectors of $S$. \\
  \end{enumerate}
  %\emph{Hint}: Rational canonical form is useful. \medskip
  
%\item Let 
%  \[
%  M=\begin{bmatrix}3&-1&0\\-1&2&-1\\0&-1&3\end{bmatrix},
%  \]
%  and let $q(x)=(x,Mx)$. Find an orthogonal matrix $P$ which
%  diagonalizes the quadratic form $q$. \\



\item 
  \begin{enumerate}
  \item Write the equation $5x_1^2-6x_1x_2+5x_2^2=1$ as $x^TAx=1$.
  \item Write $A=P^TDP$, where $D$ is a diagonal matrix and $P$
    is orthogonal with determinant $1$.
  \item Sketch the graph of the equation $x^TDx=1$ in the
  $x_1x_2$-plane.
  \item Use a geometric argument applied to part (c) to sketch the
    graph of $x^TAx=1$. \\
  \end{enumerate}


\item Repeat the previous problem for the equation $2x_1^2+6x_1x_2+2x_2^2=1$.



\end{enumerate}

\end{document}







  
