| Version 4 |
Version 3 |
| \section{Matrix Factorization} |
\section{Matrix Factorization} |
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| A \emph{matrix factorization} (or \emph{matrix decomposition}) is the right-hand-side product in |
A \emph{matrix factorization} (or \emph{matrix decomposition}) is the right-hand-side product in |
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| $$ A = F_1 F_2 \ldots F_k $$ |
$$ A = F_1 F_2 \ldots F_k $$ |
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| for ``input'' matrix $A$. The number of factor matrices $k$ depends on the situation. Most often, $k = 2$ or $k = 3$. |
for ``input'' matrix $A$. The number of factor matrices $k$ depends on the situation. Most often, $k = 2$ or $k = 3$. |
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| Note that the process of \emph{producing} a factorization/decomposition is also called ``factorization'' or ``decomposition''. |
Note that the process of \emph{producing} a factorization/decomposition is also called ``factorization'' or ``decomposition''. |
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| \section{Examples} |
\section{Examples} |
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| Some common factorizations are: |
Some common factorizations are: |
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| \begin{itemize} |
\begin{itemize} |
| \item LU-decomposition: $A = LU$, where $L$ is lower triangular, and $U$ is upper triangular |
\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 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 Singular value decomposition (SVD): $A = USV^T$, where $U$ and $V$ are orthogonal, and $S$ is a partially diagonal matrix. |
| \item The Cholesky Decomposition. |
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| \item For a positive definite matrix, we can decompose it into its \PMlinkname{square root}{SquareRootOfPositiveDefiniteMatrix} squared. |
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| \end{itemize} |
\end{itemize} |
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| 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. |
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. |
