\documentclass{urislides}

\usepackage{amsmath}

% easy way to use bold math in equation
%\def\bvec#1{\boldsymbol{#1}}
\def\bvec#1{#1}

% Beginning of document ******************************************
\begin{document}

\tinycaption{present.tex/aj02}

\begin{titlepage}
\begin{center}
{\LARGE \color{red} Important Scientific Presentation}
\end{center}
\vspace*{1.0in}
\begin{center}
Jonathan Doe \\[0.5in]
\small Department of Electrical Engineering \\
\small University of Rhode Island
\end{center}
\end{titlepage}

\begin{slide}
\slidetitle{The Initial Data Set}

\begin{itemize}
\item
Matrix \(\bvec{M}_t\) contains $c$ column vectors, \(\bvec{m}_{1}\)
through \(\bvec{m}_{c}\).
\begin{eqnarray*}
\begin{array}{cccccccc}
\bvec{M}_{t} & = & [ & \bvec{m}_{1} & \bvec{m}_{2} & 
                     \ldots & \bvec{m}_{c} & ] \\
\end{array}
\end{eqnarray*}

\item
Taking the SVD of \(\bvec{M}_t\) gives us
\begin{displaymath}
\bvec{M}_{t} = [ \bvec{\hat{U}}_{t} \hspace{10pt} \bvec{\tilde{U}}_{t} ] 
\left[ 
\begin{array}{cc}
\bvec{\hat{\Sigma}}_{t} & \bvec{0} \\
\bvec{0} & \bvec{\tilde{\Sigma}}_{t}
\end{array} 
\right]
[ \bvec{\hat{V}}_{t} \hspace{10pt} \bvec{\tilde{V}}_{t} ]^H
\end{displaymath}
where \(\bvec{\hat{U}}_{t}\) contains the $k$ left
singular vectors of \(\bvec{M}_t\) 
corresponding to its largest singular values,
which are the orthonormal basis
vectors of the desired subspace.

\end{itemize}

\end{slide}

\begin{slide}
\slidetitle{The First Iteration}

\begin{itemize}
\item
Now we create the next matrix \(\bvec{M}_{t+1}\) using the columns of
\(\bvec{M}_t\), discarding \(\bvec{m}_{1}\) and using the new
column \(\bvec{m}_{c+1}\).

\begin{eqnarray*}
\begin{array}{cccccccccc}
\bvec{M}_{t+1} & = & [ & \bvec{m}_{2} & \bvec{m}_{3} & 
                     \ldots & \bvec{m}_{c} & \bvec{m}_{c+1} & ] \\
\end{array}
\end{eqnarray*}

{\color{white}
\item
What we want are \(\bvec{\hat{U}}_{t+1}\) and
\(\bvec{\hat{\Sigma}}_{t+1}\) where
\begin{displaymath}
\bvec{M}_{t+1} = [ \bvec{\hat{U}}_{t+1} \hspace{10pt} \bvec{\tilde{U}}_{t+1} ] 
\left[ 
\begin{array}{cc}
\bvec{\hat{\Sigma}}_{t+1} & \bvec{0} \\
\bvec{0} & \bvec{\tilde{\Sigma}}_{t+1}
\end{array} 
\right]
[ \bvec{\hat{V}}_{t+1} \hspace{10pt} \bvec{\tilde{V}}_{t+1} ]^H
\end{displaymath}
}

\end{itemize}

\end{slide}

\begin{overlay}

\begin{itemize}
\item
Now we create the next matrix \(\bvec{M}_{t+1}\) using the columns of
\(\bvec{M}_t\), discarding \(\bvec{m}_{1}\) and using the new
column \(\bvec{m}_{c+1}\).

\begin{eqnarray*}
\begin{array}{cccccccccc}
\bvec{M}_{t+1} & = & [ & \bvec{m}_{2} & \bvec{m}_{3} & 
                     \ldots & \bvec{m}_{c} & \bvec{m}_{c+1} & ] \\
\end{array}
\end{eqnarray*}

{\color{black}
\item
What we want are \(\bvec{\hat{U}}_{t+1}\) and
\(\bvec{\hat{\Sigma}}_{t+1}\) where
\begin{displaymath}
\bvec{M}_{t+1} = [ \bvec{\hat{U}}_{t+1} \hspace{10pt} \bvec{\tilde{U}}_{t+1} ] 
\left[ 
\begin{array}{cc}
\bvec{\hat{\Sigma}}_{t+1} & \bvec{0} \\
\bvec{0} & \bvec{\tilde{\Sigma}}_{t+1}
\end{array} 
\right]
[ \bvec{\hat{V}}_{t+1} \hspace{10pt} \bvec{\tilde{V}}_{t+1} ]^H
\end{displaymath}
}

\end{itemize}

\end{overlay}

\begin{slide}
\slidetitle{Multiple Column Update}

\begin{itemize}
\item
More than one column can be added and removed each iteration
by adding the portion of all relevant vectors to the orthonormal
basis \(\bvec{Q}\).

\item
The matrix \(\bvec{Q}\) will be of dimension \(r \times k+2n\).

\item
The algorithm is otherwise unchanged.

\end{itemize}

\end{slide}

\end{document}