# matrix factorization

## Matrix Factorization

A matrix factorization (or 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 producing a factorization/decomposition is also called “factorization” or “decomposition”.

## Examples

Some common factorizations and related devices are:

• LU-decomposition: $A=LU$, where $L$ is lower triangular, and $U$ is upper triangular

• QR-decomposition: $A=QR$, where $Q$ is orthogonal, and $R$ is right triangular.

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

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

• Polar decomposition

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:

Title matrix factorization MatrixFactorization 2013-03-22 14:15:07 2013-03-22 14:15:07 mathcam (2727) mathcam (2727) 10 mathcam (2727) Definition msc 15A23 matrix decomposition IsawasaDecomposition factor matrix