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\pagestyle{fancy} \lhead{{\sf Homework 12 $|$ Due November 12
    (Friday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Fall 2021 $|$ M.~Macauley}}


\begin{document}

$\;$

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

\begin{enumerate}



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

\item Let $X$ be a complex inner product space, and $A\colon X\to X$.
  \begin{enumerate}
  \item Show that if $\<Ax,x\>=0$ for all $x\in X$, then $A=0$.
  \item Give an explicit example of how the previous part fails if $X$ is a
    real inner product space.
  \item Show that $A$ if self-adjoint if and only if $\<Ax,x\>\in\R$
    for all $x\in X$.
  \item Show that on a real inner product space, $A$ is self-adjoint
    and $\<Ax,x\>=0$ for all $x\in X$, then $A=0$. \bigskip
  \end{enumerate}

  %%------------------------------------------------------------------------ 
  
\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. \bigskip

%%------------------------------------------------------------------------
  
\item Fix an orthonormal basis of $\C^n$, and let $S$ be the cyclic
  shift mapping $S(a_1,\dots,a_n)=(a_2,\dots,a_n,a_1)$.
  \begin{enumerate}
  \item Prove that $S$ is unitary.
  \item Find the characteristic and minimal polynomials,
    eigenvalues, and eigenvectors of $S$.
  \item Find an orthonormal basis of $\C^n$ consisting of eigenvectors
    of $S$. \bigskip
  \end{enumerate}

%%------------------------------------------------------------------------
 
\item Consider the quadratic form $q(x)=2x_1^2+6x_1x_2+2x_2^2$. 
  \begin{enumerate}
  \item Write this as $q(x)=x^TAx$, for some $A$.
  \item Write $A=PDP^T$, where $D$ is a diagonal matrix and $P$
    is orthogonal with determinant $1$.
  \item Change variables by letting $z=P^Tx$. Sketch the level curve
    $q(x)=1$ in both the $z_1z_2$-plane and in the $x_1x_2$-plane. \bigskip
  \end{enumerate}
  
  %%------------------------------------------------------------------------  

\item Let $N\colon X\to X$ be a linear map of a finite-dimensional complex
  inner product space. Prove that the following are equivalent:
  \begin{enumerate}[(i)]
  \item $N$ is normal (that is, $NN^*=N^*N$).
  \item $N$ is unitarily similar to a diagonal matrix (i.e.,
    $N=UDU^*$).
  \item Every eigenvector of $N$ is an eigenvector of $N^*$.
  \end{enumerate}

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

\end{enumerate}

\end{document}







  
