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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Midterm II}
\date{April 9, 2003}
\maketitle

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%% \vspace*{\fill}
%% \noindent
%% For questions with an easy (E) and an advanced (A) version, choose
%% either the easy or the advanced version, not both.  You may make a
%% different choice for each question.

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%% Useful identities:
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%%   \cos \theta = \frac{e^{i\theta}+e^{-i\theta}}{2} \\
%%   \sin \theta = \frac{e^{i\theta}-e^{-i\theta}}{2i} \\
%%   \tilde f(\xi)
%%   = \frac1{\sqrt{2\pi}} \int_{-\infty}^\infty e^{-i\xi x} \, f(x) \,
%%   dx \\
%%   f(x)
%%   = \frac1{\sqrt{2\pi}} \int_{-\infty}^\infty e^{i\xi x} \,
%%   \tilde f(\xi) \, d\xi \\
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\begin{enumerate}


\item Diagonalize the matrix
\[
  A = \begin{pmatrix}
        1 & -i \\ i & 1
      \end{pmatrix} \,.
\]
I.e., find a diagonal matrix $D$ and a change of coordinate $S$ such
that $D=S^{-1}AS$.

\mbox{}\hfill (10)


\item
\begin{itemize}
\item[(E)] Are the eigenvectors of the matrix in Question 1
orthogonal?

Explain why you can answer this question without even computing the
eigenvalues. 
\hfill (4)

\item[(A)] Let $A$ be a Hermitian matrix with eigenvalues $\lambda_1,
\dots, \lambda_n$ and corresponding orthonormal eigenvectors
$\v_1,\dots,\v_n$.  Show that
\[
  A = \sum_{i=1}^n \lambda_i \, \v_i \, \v_i^H \,.
\]
\hfill (5)
\end{itemize}


\item
\begin{itemize}
\item[(E)] Find an orthonormal basis for the subspace of $\R^3$
spanned by
\[
  \v_1 = 
  \begin{pmatrix}
  1 \\ 1 \\ 1
  \end{pmatrix} \,, \qquad
  \v_2 = 
  \begin{pmatrix}
  1 \\ -1 \\ 1
  \end{pmatrix} \,.
\]
\hfill (4)

\item[(A)] Find a basis of $\R^2$ which is orthonormal with respect to
the non-standard inner product
\[
  \langle \u, \v \rangle = \u^T A \v
  \qquad \text{with }
  A = 
  \begin{pmatrix}
  2 & 1 \\ 1 & 1
  \end{pmatrix} \,.
\]
\hfill (5)
\end{itemize}


\item
\begin{itemize}
\item[(E)] Let $A$ be a Hermitian matrix, i.e.\ $A=A^H$, and consider
the standard inner product where $\langle \u,\v \rangle = \u^H \v$.
Show that
\[
  \langle \u, A\v \rangle = \langle A\u,\v \rangle \,.
\]
\hfill (4)
\item[(A)] Consider the vector space of bounded differentiable
functions with bounded first derivatives which, moverover, satisfy
$f(0)=0$.  On this vector space we define the inner product
\[
  \langle f,g \rangle
  = \int_0^\infty f^*(x) \, g(x) \, e^{-x} \, dx \,.
\]
Show that the operator
\[
  \mathcal{L}f
  = e^x \, \frac{d}{dx} \left( e^{-x} \, \frac{df}{dx} \right)
\]
is Hermitian, i.e.\ that $\langle f, \mathcal{L}g \rangle =\langle
\mathcal{L}f, g \rangle$.
\hfill (5)
\end{itemize}


\item Show that if $f$ is an even, real-valued function, i.e.\ if
$f(x)=f(-x)$ and $f^*(x)=f(x)$, then its Fourier transform is a
real-valued function as well.  \hfill (5)
\item 
\begin{itemize}
\item[(E)] Show that $\mathcal{F}(f(x+a)) = e^{i\xi a} \,
\mathcal{F}(f)$, where $\mathcal{F}(f)=\tilde f$ denotes the Fourier
transform of $f$.  \hfill (4)
\item[(A)] Show that if $f$ is periodic with period $a$, then $\tilde
f(\xi)=0$ unless $\xi a=2\pi n$ for some integer $n$.
\hfill (5)
\end{itemize}


\item Compute
\[
  \int_{-\infty}^\infty \delta(e^{2x}-1) \, e^x \, dx \,.
\]
\hfill (5)


\item
\begin{itemize}
\item[(E)] Show that the Fourier transform of
\[
  f(x) = 
  \begin{cases}
  0 & \text{for } x<0 \\
  e^{-x} \, \sin x & \text{for } x \geq 0
  \end{cases} \,.
\]
is
\[
  \tilde f (\xi)
  = \frac1{\sqrt{2\pi}} \, \frac1{2i} \,
    \left( \frac1{1-i+i \xi} - \frac1{1+i+i \xi} \right) \,.
\]

\hfill (8)
\item[(A)] Prove that
\[
  \int_0^\infty e^{-2x} \, \sin^2 x \, dx
  = \frac1\pi \int_0^\infty \frac1{4 + \xi^4} \, d \xi \,.
\]
Hint: You may use the result from part (A) without proof.  \hfill (10)
\end{itemize}


\end{enumerate}

\end{document}
