matrix factorizationMatrixFactorization2004-03-12 23:43:002007-06-16 22:50:36Definitionfactor matrix\usepackage{amssymb}
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A \emph{matrix factorization} (or \emph{matrix decomposition}) is the right-hand-side product in
$$ A = F_1 F_2 \ldots F_k $$
for ``input'' matrix $A$. The number of factor matrices $k$ depends on the situation. Most often, $k = 2$ or $k = 3$.
Note that the process of \emph{producing} a factorization/decomposition is also called ``factorization'' or ``decomposition''.
\subsubsection*{Examples}
Some common factorizations and related devices are:
\begin{itemize}
\item LU-decomposition: $A = LU$, where $L$ is lower triangular, and $U$ is upper triangular
\item QR-decomposition: $A = QR$, where $Q$ is orthogonal, and $R$ is right triangular.
\item Singular value decomposition (SVD): $A = USV^T$, where $U$ and $V$ are orthogonal, and $S$ is a partially diagonal matrix.
\item The Cholesky Decomposition.
\item For a positive definite matrix, we can decompose it into its \PMlinkname{square root}{SquareRootOfPositiveDefiniteMatrix} squared.
\item Polar decomposition
\item Jordan canonical form
\item Iwasawa decomposition
\end{itemize}
See the entries for these and other matrix factorizations for details on the contents of the factor matrices, where to apply them, and how to best calculate them.
\subsubsection*{Simultaneous matrix factorization}
A related problem is to diagonalize or tridiagonalize many matrices using
the same matrix. Some results in this direction are listed below:
\begin{itemize}
\item commuting matrices are simultanenously triangularizable
\item commuting normal matrices are simultanenously diagonalizable
\end{itemize}