Matrix decomposition

In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.

In numerical analysis, different decompositions are used to implement efficient matrix algorithms.

For instance, when solving a system of linear equations ${\displaystyle Ax=b}$, the matrix A can be decomposed via the LU decomposition. The LU decomposition factorizes a matrix into a lower triangular matrix L and an upper triangular matrix U. The systems ${\displaystyle L(Ux)=b}$ and ${\displaystyle Ux=L^{-1}b}$ require fewer additions and multiplications to solve, compared with the original system ${\displaystyle Ax=b}$, though one might require significantly more digits in inexact arithmetic such as floating point.

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