matrix factorization
Matrix Factorization
A matrix factorization (or matrix decomposition) is the righthandside product in
$$A={F}_{1}{F}_{2}\mathrm{\dots}{F}_{k}$$ 
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 producing a factorization/decomposition is also called “factorization” or “decomposition”.
Examples
Some common factorizations and related devices are:

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LUdecomposition: $A=LU$, where $L$ is lower triangular, and $U$ is upper triangular

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QRdecomposition: $A=QR$, where $Q$ is orthogonal^{}, and $R$ is right triangular.

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Singular value decomposition^{} (SVD): $A=US{V}^{T}$, where $U$ and $V$ are orthogonal, and $S$ is a partially diagonal matrix^{}.

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The Cholesky Decomposition^{}.

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For a positive definite matrix, we can decompose it into its square root (http://planetmath.org/SquareRootOfPositiveDefiniteMatrix) squared.

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Polar decomposition
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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.
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:

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commuting matrices^{} are simultanenously triangularizable

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commuting normal matrices^{} are simultanenously diagonalizable
Title  matrix factorization 

Canonical name  MatrixFactorization 
Date of creation  20130322 14:15:07 
Last modified on  20130322 14:15:07 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  10 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 15A23 
Synonym  matrix decomposition 
Related topic  IsawasaDecomposition 
Defines  factor matrix 