\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig, amsmath, latexsym, enumerate}
\usepackage[all]{xy}
\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=0in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in

\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\def\<{\langle}
\def\>{\rangle}
\DeclareMathOperator{\im}{Im}
\DeclareMathOperator{\Span}{span}
\DeclareMathOperator{\Char}{char}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\tr}{tr}

\newcommand{\vv}[2]{\begin{bmatrix} #1 \\ #2 \end{bmatrix}}
\newcommand{\vvv}[3]{\begin{bmatrix} #1 \\ #2 \\ #3 \end{bmatrix}}
\newcommand{\vvvv}[4]{\begin{bmatrix} #1 \\ #2 \\ #3 \\ #4 \end{bmatrix}}

\renewcommand{\footrulewidth}{1pt}
\newenvironment{proof}[1][Proof.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{sol}[1][Solution.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{fminipage}%
{\setlength{\fboxsep}{15pt}\begin{Sbox}\begin{minipage}}%
{\end{minipage}\end{Sbox}\fbox{\TheSbox}}
\newcommand{\qed}{\hfill \ensuremath{\Box}}
\newcommand{\disp}{\displaystyle}
\newcommand{\ds}{\displaystyle}


\pagestyle{fancy} \lhead{{\sf Homework 13 $|$ Due December 6 (Monday)
}} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear Algebra $|$
    Fall 2020 $|$ M.~Macauley}}


\begin{document}

$\;$

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

\begin{enumerate}
  
%%------------------------------------------------------------------------

%\item Let $\lambda$ be an eigenvalue of $A\colon X\to X$ with
%  $|\lambda|$ maximized.
%  \begin{enumerate}
%  \item Show that $||A||\geq |\lambda|$
%  \item Give an example of an $A$ where equality does not hold.
%  \item Show that if $A$ is normal, then equality holds. 
%  \end{enumerate}
  %Prove that $||N||=\max|n_i|$, where the $n_i$s are the
  %eigenvalues of $N$. \\

%%------------------------------------------------------------------------
  
\item For any positive-definite self-adjoint mapping $M\colon X\to X$,
  define an inner product on $X$ by $\<x,y\>:=(x,My)$. Throughout this
  problem, let $X=\R^2$ and $M=\begin{bmatrix}2 & -1 \\ -1 &
  2\end{bmatrix}$.
  \begin{enumerate}
  \item Find two orthonormal bases for $X$ that contain
    the vector $e_1/||e_1||$, where $e_1=(1,0)$.
  \item Find two orthonormal bases for $X$ that contain
    the vector $e_2/||e_2||$, where $e_2=(0,1)$.
  \item Find an vector $v_2$ orthogonal to $v_1=(1,1)$.
  \item Find a matrix $H$ that is self-adjoint with respect to $(\;,\;)$, but
    \emph{not} with respect to $\<\;,\;\>$. \\
  \end{enumerate}

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

\item Let $H,M\colon X\to X$ be self-adjoint mappings, and $M$
  positive-definite. 
  \begin{enumerate}
  \item Formulate and prove a necessary and sufficient condition for
    $M^{-1}H$ to be self-adjoint with respect to the standard inner
    product.
  \item Prove that $M^{-1}H$ is self-adjoint with respect to the inner
    product $\<x,y\>=(x,My)$.
  \item Prove that if $H$ is positive-definite, then so is $M^{-1}H$. \\
  \end{enumerate}

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

\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 $\mu=\min\{R_{H,M}(x)\mid x\in X\}$. Show that $\mu$
    exists, and that the $v\in X$ for which $R_{H,M}(v)=\mu$ satisfies
    $Hv=\mu Mv$.
  \item Show that the constrained minimum problem
    \[
    \min\big\{R_{H,M}(x)\mid (x,Mv)=0\big\}\,
    \]
    has a nonzero solution $w\in X$, which satisfies $Hw=\kappa Mw$,
    where $\kappa=R_{H,M}(w)$. \\
  \end{enumerate}

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

\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
    \[
    Hv_i=\mu_iM v_i\quad \mbox{($\mu_i$ real)},\qquad
    (v_i,Mv_j)=\left\{\begin{array}{cc}1&i=j\\0&i\neq j\end{array}\right.
    \]
  \item Compute $(v_i,Hv_j)$, and show that there is an invertible
    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}

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

\end{enumerate}

\end{document}







  
