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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 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 L(Ux)=b and Ux=L^{-1}b require fewer additions and multiplications to solve, compared with the original system Ax=b, though one might require significantly more digits in inexact arithmetic such as floating point.
Similarly, the QR decomposition expresses A as QR with Q an orthogonal matrix and R an upper triangular matrix. The system Q(Rx) = b is solved by Rx = Q^{T}b = c, and the system Rx = c is solved by 'back substitution'. The number of additions and multiplications required is about twice that of using the LU solver, but no more digits are required in inexact arithmetic because the QR decomposition is numerically stable.
The Jordan normal form and the Jordan–Chevalley decomposition
There exist analogues of the SVD, QR, LU and Cholesky factorizations for quasimatrices and cmatrices or continuous matrices.^{[7]} A ‘quasimatrix’ is, like a matrix, a rectangular scheme whose elements are indexed, but one discrete index is replaced by a continuous index. Likewise, a ‘cmatrix’, is continuous in both indices. As an example of a cmatrix, one can think of the kernel of an integral operator.
These factorizations are based on early work by Fredholm (1903), Hilbert (1904) and Schmidt (1907). For an account, and a translation to English of the seminal papers, see Stewart (2011).
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