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Representation of a matrix as a product
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 , 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 and require fewer additions and multiplications to solve, compared with the original system , 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.

Traditionally applicable to: square matrix A, although rectangular matrices can be applicable. Decomposition: , where L is lower triangular and U is upper triangular. Related: the LDU decomposition is , where L is lower triangular with ones on the diagonal, U is upper triangular with ones on the diagonal, and D is a diagonal matrix. Related: the LUP decomposition is , where L is lower triangular, U is upper triangular, and P is a permutation matrix. Existence: An LUP decomposition exists for any square matrix A. When P is an identity matrix, the LUP decomposition reduces to the LU decomposition. Comments: The LUP and LU decompositions are useful in solving an n by n system of linear equations These decompositions summarize the process of Gaussian elimination in matrix form. Matrix P represents any row interchanges carried out in the process of Gaussian elimination. If Gaussian elimination produces the row echelon form without requiring any row interchanges, then P = I, so an LU decomposition exists.

Applicable to: square, complex, symmetric matrix A.
Decomposition: , where D is a real nonnegative diagonal matrix, and V is unitary. denotes the matrix transpose of V.
Comment: The diagonal elements of D are the nonnegative square roots of the eigenvalues of Comment: V may be complex even if A is real. Comment: This is not a special case of the eigendecomposition (see above), which uses instead of Moreover, if A is not real, it is not Hermitian and the form using also does not apply.

Applicable to: m by n matrix A.
Decomposition: , where D is a nonnegative diagonal matrix, and U and V satisfy Here is the conjugate transpose of V (or simply the transpose, if V contains real numbers only), and I denotes the identity matrix (of some dimension). Comment: The diagonal elements of D are called the singular values of A. Comment: Like the eigendecomposition above, the singular value decomposition involves finding basis directions along which matrix multiplication is equivalent to scalar multiplication, but it has greater generality since the matrix under consideration need not be square. Uniqueness: the singular values of are always uniquely determined. and need not to be unique in general.

Refers to variants of existing matrix decompositions, such as the SVD, that are invariant with respect to diagonal scaling. Applicable to: m by n matrix A. Unit Scale Invariant Singular Value Decomposition: , where S is a unique nonnegative diagonal matrix of scale invariant singular values, U and V are unitary matrices, is the conjugate transpose of V, and positive diagonal matrices D and E.
Comment: Is analogous to the SVD except that the diagonal elements of S are invariant with respect to left and/or right multiplication of A by arbitrary nonsingular diagonal matrices, as opposed to the standard SVD for which the singular values are invariant with respect to left and/or right multiplication of A by arbitrary unitary matrices. Comment: Is an alternative to the standard SVD when invariance is required with respect to diagonal rather than unitary transformations of A.
Uniqueness: The scale invariant singular values of (given by the diagonal elements of S) are always uniquely determined. Diagonal matrices D and E, and unitary U and V, are not necessarily unique in general. Comment: U and V matrices are not the same as those from the SVD. Analogous scale invariant decompositions can be derived from other matrix decompositions; for example, to obtain scale invariant eigenvalues.

Applicable to: square matrix A.
Decomposition: where is the Hessenberg matrix and is a unitary matrix. Comment: often the first step in the Schur decomposition.

Also known as: UTV decomposition, ULV decomposition, URV decomposition. Applicable to: m by n matrix A.
Decomposition: , where T is a triangular matrix, and U and V are unitary matrices. Comment: Similar to the singular value decomposition and to the Schur decomposition.