Funes el memorioso

\documentclass[10pt]{article}

% for smallper page
\usepackage[papersize={4.2in, 1.4in}, text={4in, 1.3in}]{geometry}
% for fancy math
\usepackage{amsmath}

% rank operator
\DeclareMathOperator*{\rank}{rank}

% Matrix transpose
\newcommand{\trans}[1]{\ensuremath{{#1}^\top}}

% for extra space at the end of abbreviation
\usepackage{xspace}

% positive semi-definite
\newcommand{\psd}{\textsc{psd}\xspace}

% boldface uppercase letters for matrices
\newcommand{\Abf}{\ensuremath{\mathbf A}}
\newcommand{\Bbf}{\ensuremath{\mathbf B}}

% boldface lowercase letters for vectors
\newcommand{\xbf}{\ensuremath{\mathbf x}}

% for math blackboard font
\usepackage{amssymb}
% set of real numbers
\newcommand{\Rbb}{\ensuremath{\mathbb R}}

\usepackage{palatino}
\usepackage[sc]{mathpazo}

\begin{document}
\noindent
For any real symmetric matrcies $\Abf$ such that $\rank(\Abf_{n\times n})=r$,
the following statements are equivalent
and any one of them can serve as the definition of
\emph{positive semi-definite} (\psd) matrices.
\begin{itemize}
  \item $\trans\xbf \Abf\xbf \geq 0$ for any non-zero vector
    $\xbf\in\Rbb^{n\times 1}$.
    \hfill\refstepcounter{equation}\textup{(\theequation)}%
  \item All the $n$ eigenvalues of $\Abf$ are non-negative.
    \hfill\refstepcounter{equation}\textup{(\theequation)}%
  \item $\Abf=\trans\Bbf \Bbf$ for some $\Bbf$ with $\rank(\Bbf)=r$.
    \hfill\refstepcounter{equation}\textup{(\theequation)}%
\end{itemize}

\end{document}

\documentclass{beamer}
\usepackage{hyperref}
\setbeamertemplate{theorems}[numbered]
\usetheme{CambridgeUS}

\begin{document}
\begin{frame}
  \frametitle{Test}
  \begin{theorem}[Fermat's Last Theorem]
    $x^n + y^n = z^n$.
    \label{thm:fermat}
  \end{theorem}
  As shown in Theorem \ref{thm:fermat}...

  \begin{example}
    Here goes an example!
  \end{example}

  \begin{theorem}
    $e^{i\pi} + 1 = 0$
  \end{theorem}

  \begin{corollary}
    Corollary!
  \end{corollary}
\end{frame}

\end{document}