In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. There are several different decompositions of a given matrix and the decomposition used depends on the problem we want to solve as well as the matrix to be factorized. In numerical analysis for example different decompositions are used to implement efficient matrix algorithms.

Example

When solving a system of linear equations 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 matrices L and U are much easier to solve than the original matrix A.

Common decompositions

• Block LU decomposition
• Cholesky decomposition
• Jordan decomposition
• LU decomposition
• Polar decomposition
• Proper orthogonal decomposition
• QR decomposition
• Schur decomposition
• Singular value decomposition
• Spectral decomposition (also called the eigendecomposition)

Matrix theory