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\pagestyle{fancy} \lhead{{\sf Homework 12 $|$ Due April 20 (Monday)
}} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear Algebra $|$
    Spring 2015 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Lax, Chapter 8, pages 101--120. \\

\begin{enumerate}
  
\item 
  \begin{enumerate}
  \item Write the equation $5x_1^2-6x_1x_2+5x_2^2=1$ in the form
    $x^TAx=1$.
  \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 $x^TAx=1$.
  \item Repeat Parts (a)--(d) for the equation
    $2x_1^2+6x_1x_2+2x_2^2=1$.
  \end{enumerate}

  

  % 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 an isometry in the Euclidean norm. 
  \item Determine the eigenvalues and eigenvectors of $S$.
  \item Verify that the eigenvectors are orthogonal.
  \end{enumerate}
  \emph{Hint}: There are very short and elegant solutions for all three
  parts of this problem! You may find the last problem on HW 9 useful.



  % Lax Exercise 8.10 (p. 119)
\item Let $N\colon X\to X$ be a normal mapping of a Euclidean
  space. Prove that $||N||=\max|n_i|$, where the $n_i$s are the
  eigenvalues of $N$.
  
  


  % Lax Exercise 8.5 (p. 118)
\item Let $H,M\colon X\to X$ be self-adjoint mappings, and $M$ positive
  definite. Define
  \[
  R_{H,M}(x)=\frac{(x,Hx)}{(x,Mx)}\,.
  \]
  \begin{enumerate}
  \item Let $\mu=\inf\{R_{H,M}(x)\mid x\in X\}$. Show that $\mu$
    exists, and that there is some $v\in X$ for which
    $R_{H,M}(v)=\mu$, and that $\mu$ and $v$ satisfy $Hv=\mu Mv$.
  \item Show that the constrained minimum problem
    \[
    \min\{R_{H,M}(y)\mid (y,Mv)=0\}\,
    \]
    has a nonzero solution $w\in X$, and that this solution satisfies
    $Hw=\kappa Mw$, where $\kappa=R_{H,M}(w)$.
  \end{enumerate}



% Lax Exercise 8.6-8.7 (p. 118)
\item Let $H,M\colon X\to X$ be self-adjoint mappings, and $M$
  positive definite.
  \begin{enumerate}
  \item Show that there exists a basis $v_1,\dots,v_n$ of $X$ where
    each $v_i$ satisfies an equation of the form
    \[
    Hv_i=\mu_iM v_i\quad \mbox{($\mu_i$ real)},\qquad
    (v_i,Mv_j)=\left\{\begin{array}{cc}1&i=j\\0&i\neq j\end{array}\right.
    \]
  \item Compute $(v_i,Hv_j)$, and show that there is an invertible
    real matrix $U$ for which $U^*MU=I$ and $U^*HU$ is diagonal.
  \item Characterize the numbers $\mu_1,\dots\mu_n$ by a minimax
    principle.
  \end{enumerate}



% Lax Exercise 8.8-9 (p. 119)
\item Let $H,M\colon X\to X$ be self-adjoint mappings, and $M$
  positive definite. 
  \begin{enumerate}
    \item Prove that all the eigenvalues of $M^{-1}H$ are real.
    \item Prove that if $H$ is positive-definite, then all the
      eigenvalues of $M^{-1}H$ are positive.
    \item Show by example how Part~(b) can fail if $M$ is not positive
      definite.
  \end{enumerate}


\end{enumerate}

\end{document}







  
