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%  LaTeX file # 1 for ``learning Latex by example''
%  A ``bare-bones'' example. It illustrates the use of:
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%   $ $ to enclose math symbols in a line 
%       (as opposed to in ``display math'' mode).
%       e.g., ``an eigenvalue $\lambda$''.
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\documentstyle[12pt]{article}
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\begin{document}

\begin{center} 
   {\bf Sample \LaTeX file } 
\end{center}

\vspace{5mm}
\noindent{\sc Problem:} Show that the eigenvalues of self-adjoint 
matrix $A$ are real.

\vspace{5mm}
\noindent{\sc Solution:}
  Let $\lambda$ be an eigenvalue of $A$, with corresponding eigenvector $u$ 
with $\Vert u \Vert=1$ (here, $\Vert\cdot\Vert$ denotes the $2$-norm).
Since $Au=\lambda u$, we have 
\[
   u^HAu = \lambda, 
\]
where $\cdot^H$ denotes conjugate transpose. Now, 
since $\lambda$ is a scalar, we have $\lambda^H = \bar{\lambda}$ 
(the ``bar'' denotes complex conjugation), so that
\begin{eqnarray*}
  \bar{\lambda} = \lambda^H &=& (u^HAu)^H \\
                            &=& u^HA^H u \\
                            &=& u^HAu \\
                            &=& \lambda.
\end{eqnarray*}
Hence $\bar{\lambda}=\lambda$, so that $\lambda$ must be real.

Incidentally, the ``trace'' of a matrix $A$ is defined to be
\[
   Tr(A) = \sum_{i=1}^n \lambda_i, 
\]
which I include here simply to demonstrate how to use
the summation notation $\sum$ in \LaTeX.

\end{document}