% formatted on 970304 by morris
\documentstyle{article}
\setcounter{page}{45}
\textheight 9.5in
\textwidth 5.75in
\voffset -1in
\hoffset -0.55in
%\documentstyle{llncs}
%\pagestyle{empty}
\begin{document}
\title{Orthogonal matrix decompositions of \\ matrix sequences 
and applications \\}
      
\author{Paul Van Dooren \\
Universit{\'e} Catholique de Louvain,\\
Dept. Mathematical Engineering, \\
B-1348, Louvain-la-Neuve, Belgium \\
}
\date{}

\maketitle
In this lecture we survey orthogonal matrix decompositions of matrix
sequences and their applications.  The three basic matrix
decompositions involving orthogonal transformations are the $QR$
decomposition, the {\it Schur} decomposition, and the {\it singular
value} decomposition of a matrix $A$.  In the last two decades these
decompositions have been extended to the case where $A$ is not just a
single matrix but a product or quotient of two or more matrices $A_i$.
These extensions include
\begin{itemize}
\item the generalized $QR$ decomposition of two or more matrices
\item the periodic Schur form for standard and generalized eigenvalue
problems
\item the product and quotient $SVD$ of two or more matrices.
\end{itemize}

We discuss the definitions, properties and algorithms for these
decompositions and we also show their use in several basic problems
from various applications.

\bigskip

{\bf References}

\medskip

[1] B. De Moor, P. Van Dooren, ``Generalizations of the singular
value and QR decompositions'', SIMAX, Vol. 13:4, pp. 993-1014, Oct. 1992.

\smallskip

[2] A. Bojanczyk, G. Golub, P. Van Dooren, ``The periodic Schur
decomposition.  Algorithms and applications'', in Advanced Signal
Processing Algorithms, Architectures, and Implementations III,
Proceedings of SPIE, the International Society for Optical
Engineering, Vol. 1770, pp. 31-42, Ed. F.T. Luk, Bellingham, Wash.,
USA, July 1992.

\smallskip

[3] G. Golub, K. Solna, P. Van Dooren, ``Computing the singular
values of products and quotients of matrices,'' pp. 139-147, in
SVD in Signal Processing III, Algorithms, Architectures and Applications,
Eds. M. Moonen, B. De Moor, Elsevier, Amsterdam, 1995


\end{document}

