# 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 , 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.

### Cholesky decomposition

• Applicable to: square, hermitian, positive definite matrix A
• Decomposition: , where U is upper triangular with real positive diagonal entries
• Comment: if the matrix A is Hermitian and positive semi-definite, then it has a decomposition of the form if the diagonal entries of are allowed to be zero
• Uniqueness: for positive definite matrices Cholesky decomposition is unique. However, it is not unique in the positive semi-definite case.
• Comment: if A is real and symmetric, has all real elements
• Comment: An alternative is the LDL decomposition, which can avoid extracting square roots.

### QR decomposition

• Applicable to: m-by-n matrix A
• Decomposition: where Q is a unitary matrix of size m-by-m, and R is an upper triangular matrix of size m-by-n
• Uniqueness: In general it is not unique, but if is of full rank, then there exists a single that has all positive diagonal elements. If is square, also is unique.
• Comment: The QR decomposition provides an alternative way of solving the system of equations without inverting the matrix A. The fact that Q is orthogonal means that , so that is equivalent to , which is easier to solve since R is triangular.

### Eigendecomposition

• Also called spectral decomposition.
• Applicable to: square matrix A with linearly independent eigenvectors (not necessarily distinct eigenvalues).
• Decomposition: , where D is a diagonal matrix formed from the eigenvalues of A, and the columns of V are the corresponding eigenvectors of A.
• Existence: An n-by-n matrix A always has n (complex) eigenvalues, which can be ordered (in more than one way) to form an n-by-n diagonal matrix D and a corresponding matrix of nonzero columns V that satisfies the eigenvalue equation . is invertible if and only if the n eigenvectors are linearly independent (i.e., each eigenvalue has geometric multiplicity equal to its algebraic multiplicity). A sufficient (but not necessary) condition for this to happen is that all the eigenvalues are different (in this case geometric and algebraic multiplicity are equal to 1)
• Comment: One can always normalize the eigenvectors to have length one (see the definition of the eigenvalue equation)
• Comment: Every normal matrix A (i.e., matrix for which , where is a conjugate transpose) can be eigendecomposed. For a normal matrix A (and only for a normal matrix), the eigenvectors can also be made orthonormal ( ) and the eigendecomposition reads as . In particular all unitary, Hermitian, or skew-Hermitian (in the real-valued case, all orthogonal, symmetric, or skew-symmetric, respectively) matrices are normal and therefore possess this property.
• Comment: For any real symmetric matrix A, the eigendecomposition always exists and can be written as , where both D and V are real-valued.
• Comment: The eigendecomposition is useful for understanding the solution of a system of linear ordinary differential equations or linear difference equations. For example, the difference equation starting from the initial condition is solved by , which is equivalent to , where V and D are the matrices formed from the eigenvectors and eigenvalues of A. Since D is diagonal, raising it to power , just involves raising each element on the diagonal to the power t. This is much easier to do and understand than raising A to power t, since A is usually not diagonal.

### 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.

### Real Schur decomposition

• Applicable to: square matrix A
• Decomposition: This is a version of Schur decomposition where and only contain real numbers. One can always write where V is a real orthogonal matrix, is the transpose of V, and S is a block upper triangular matrix called the real Schur form. The blocks on the diagonal of S are of size 1×1 (in which case they represent real eigenvalues) or 2×2 (in which case they are derived from complex conjugate eigenvalue pairs).

### QZ decomposition

• Also called: generalized Schur decomposition
• Applicable to: square matrices A and B
• Comment: there are two versions of this decomposition: complex and real.
• Decomposition (complex version): and where Q and Z are unitary matrices, the H superscript represents conjugate transpose, and S and T are upper triangular matrices.
• Comment: in the complex QZ decomposition, the ratios of the diagonal elements of S to the corresponding diagonal elements of T, , are the generalized eigenvalues that solve the generalized eigenvalue problem (where is an unknown scalar and v is an unknown nonzero vector).
• Decomposition (real version): and where A, B, Q, Z, S, and T are matrices containing real numbers only. In this case Q and Z are orthogonal matrices, the T superscript represents transposition, and S and T are block upper triangular matrices. The blocks on the diagonal of S and T are of size 1×1 or 2×2.

### Takagi's factorization

• 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 .

### Singular value decomposition

• Applicable to: m-by-n matrix A.
• Decomposition: , where D is a nonnegative diagonal matrix, and U and V are unitary matrices, and is the conjugate transpose of V (or simply the transpose, if V contains real numbers only).
• 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.

### Scale-invariant decompositions

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, e.g., to obtain scale-invariant eigenvalues.[3][4]

## Other decompositions

### Polar decomposition

• Applicable to: any square complex matrix A.
• Decomposition: (right polar decomposition) or (left polar decomposition), where U is a unitary matrix and P and P' are positive semidefinite Hermitian matrices.
• Uniqueness: is always unique and equal to (which is always hermitian and positive semidefinite). If is invertible, then is unique.
• Comment: Since any Hermitian matrix admits a spectral decomposition with a unitary matrix, can be written as . Since is positive semidefinite, all elements in are non-negative. Since the product of two unitary matrices is unitary, taking one can write which is the singular value decomposition. Hence, the existence of the polar decomposition is equivalent to the existence of the singular value decomposition.

### Algebraic polar decomposition

• Applicable to: square, complex, non-singular matrix A.[5]
• Decomposition: , where Q is a complex orthogonal matrix and S is complex symmetric matrix.
• Uniqueness: If has no negative real eigenvalues, then the decomposition is unique.[6]
• Comment: The existence of this decomposition is equivalent to being similar to .[7]
• Comment: A variant of this decomposition is , where R is a real matrix and C is a circular matrix.[6]

### Mostow's decomposition

• Applicable to: square, complex, non-singular matrix A.[8][9]
• Decomposition: , where U is unitary, M is real anti-symmetric and S is real symmetric.
• Comment: The matrix A can also be decomposed as , where U2 is unitary, M2 is real anti-symmetric and S2 is real symmetric.[6]

### Sinkhorn normal form

• Applicable to: square real matrix A with strictly positive elements.
• Decomposition: , where S is doubly stochastic and D1 and D2 are real diagonal matrices with strictly positive elements.

### Sectoral decomposition[10]

• Applicable to: square, complex matrix A with numerical range contained in the sector .
• Decomposition: , where C is an invertible complex matrix and with all .[10][11]

## Generalizations

There exist analogues of the SVD, QR, LU and Cholesky factorizations for quasimatrices and cmatrices or continuous matrices.[13] 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).

## Notes

1. Simon & Blume 1994 Chapter 7.
2. Piziak, R.; Odell, P. L. (1 June 1999). "Full Rank Factorization of Matrices". Mathematics Magazine. 72 (3): 193. doi:10.2307/2690882.
3. Uhlmann, J.K. (2018), A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations, SIAM Journal on Matrix Analysis, 239:2, pp. 781–800
4. Uhlmann, J.K. (2018), A Rank-Preserving Generalized Matrix Inverse for Consistency with Respect to Similarity, IEEE Control Systems Letters, ISSN 2475-1456
5. Choudhury & Horn 1987, pp. 219–225
6. Bhatia, Rajendra (2013-11-15). "The bipolar decomposition". Linear Algebra and its Applications. 439 (10): 3031–3037. doi:10.1016/j.laa.2013.09.006.
7. Horn & merino 1995, pp. 43–92
8. Mostow, G. D. (1955), Some new decomposition theorems for semi-simple groups, Mem. Amer. Math. Soc., 14, pp. 31–54
9. Nielsen, Frank; Bhatia, Rajendra (2012). Matrix Information Geometry. Springer. p. 224. doi:10.1007/978-3-642-30232-9. ISBN 9783642302329.
10. Zhang, Fuzhen (30 June 2014). "A matrix decomposition and its applications". Linear and Multilinear Algebra: 1–10. doi:10.1080/03081087.2014.933219.
11. 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.
12. Idel, Martin; Soto Gaona, Sebastián; Wolf, Michael M. (2017-07-15). "Perturbation bounds for Williamson's symplectic normal form". Linear Algebra and its Applications. 525: 45–58. arXiv:1609.01338. doi:10.1016/j.laa.2017.03.013.
13. Townsend & Trefethen 2015

## References

• Choudhury, Dipa; Horn, Roger A. (April 1987). "A Complex Orthogonal-Symmetric Analog of the Polar Decomposition". SIAM Journal on Algebraic Discrete Methods. 8 (2). doi:10.1137/0608019.
• Fredholm, I. (1903), "Sur une classe d'´equations fonctionnelles", Acta Mathematica (in French), 27: 365–390, doi:10.1007/bf02421317
• Hilbert, D. (1904), "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen", Nachr. Königl. Ges. Gött (in German), 1904: 49–91
• Horn, Roger A.; Merino, Dennis I. (January 1995). "Contragredient equivalence: A canonical form and some applications". Linear Algebra and its Applications. 214. doi:10.1016/0024-3795(93)00056-6.
• Meyer, C. D. (2000), Matrix Analysis and Applied Linear Algebra, SIAM, ISBN 978-0-89871-454-8
• Schmidt, E. (1907), "Zur Theorie der linearen und nichtlinearen Integralgleichungen. I Teil. Entwicklung willkürlichen Funktionen nach System vorgeschriebener", Mathematische Annalen (in German), 63: 433–476, doi:10.1007/bf01449770
• Simon, C.; Blume, L. (1994). Mathematics for Economists. Norton. ISBN 0-393-95733-0.
• Stewart, G. W. (2011), Fredholm, Hilbert, Schmidt: three fundamental papers on integral equations (PDF), retrieved 2015-01-06
• Townsend, A.; Trefethen, L. N. (2015), "Continuous analogues of matrix factorizations", Proc. R. Soc. A, 471 (2173), Bibcode:2014RSPSA.47140585T, doi:10.1098/rspa.2014.0585, PMC 4277194