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.

Example

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 = QTb = 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.

Decompositions related to solving systems of linear equations

LU decomposition

Main article: LU decomposition

LU reduction

Main article: LU reduction

Block LU decomposition

Rank factorization

Main article: Rank factorization

Cholesky decomposition

QR decomposition

Main article: QR decomposition

RRQR factorization

Main article: RRQR factorization

Interpolative decomposition

Decompositions based on eigenvalues and related concepts

Eigendecomposition

Jordan decomposition

The Jordan normal form and the Jordan–Chevalley decomposition

Schur decomposition

Main article: Schur decomposition

QZ decomposition

Main article: QZ decomposition

Takagi's factorization

Singular value decomposition

Other decompositions

Polar decomposition

Main article: Polar decomposition

Algebraic polar decomposition

Sinkhorn normal form

Main article: Sinkhorn's theorem

Sectoral decomposition[5]

Generalizations

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).

See also

Notes

  1. Simon & Blume 1994 Chapter 7.
  2. Meyer 2000, p. 514
  3. Choudhury & Horn 1987, pp. 219–225
  4. Horn & merino 1995, pp. 43–92
  5. 1 2 Zhang, Fuzhen (30 June 2014). "A matrix decomposition and its applications". Linear and Multilinear Algebra: 1–10. doi:10.1080/03081087.2014.933219.
  6. Drury, S.W. (November 2013). "Fischer determinantal inequalities and Highamʼs Conjecture". Linear Algebra and its Applications 439 (10): 3129–3133. doi:10.1016/j.laa.2013.08.031.
  7. Townsend & Trefethen 2015

References

External links

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