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 ${\displaystyle Ax=b}$, 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.

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

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LU reduction

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Block LU decomposition

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Rank factorization

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Cholesky decomposition

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QR decomposition

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RRQR factorization

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Interpolative decomposition

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Decompositions based on eigenvalues and related concepts

Eigendecomposition

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Jordan decomposition

• Applicable to: square matrix A
• Comment: the Jordan normal form generalizes the eigendecomposition to cases where there are repeated eigenvalues and cannot be diagonalized, the Jordan–Chevalley decomposition does this without choosing a basis.

Schur decomposition

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QZ decomposition

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Singular value decomposition

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Other decompositions

Polar decomposition

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Generalizations

Template:Expand section There exist analogues of the SVD, QR, LU and Cholesky factorizations for quasimatrices and cmatrices or continuous matrices.[5] 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 Template:Harvtxt, Template:Harvtxt and Template:Harvtxt. For an account, and a translation to English of the seminal papers, see Template:Harvtxt.

References

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