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 edit

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.

Decompositions related to solving systems of linear equations edit

LU decomposition edit

LU reduction edit

Block LU decomposition edit

Rank factorization edit

Cholesky decomposition edit

  • Applicable to: square, hermitian, positive definite matrix  
  • Decomposition:  , where   is upper triangular with real positive diagonal entries
  • Comment: if the matrix   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   is real and symmetric,   has all real elements
  • Comment: An alternative is the LDL decomposition, which can avoid extracting square roots.

QR decomposition edit

  • Applicable to: m-by-n matrix A with linearly independent columns
  • Decomposition:   where   is a unitary matrix of size m-by-m, and   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 effective way to solve the system of equations  . The fact that   is orthogonal means that  , so that   is equivalent to  , which is very easy to solve since   is triangular.

RRQR factorization edit

Interpolative decomposition edit

Decompositions based on eigenvalues and related concepts edit

Eigendecomposition edit

  • 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 (that is, 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 (that is, 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 edit

The Jordan normal form and the Jordan–Chevalley 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 edit

Real Schur decomposition edit

  • 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 edit

  • 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 * 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 edit

  • 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.

Singular value decomposition edit

  • 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.

Scale-invariant decompositions edit

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.[3][4]

Hessenberg decomposition edit

  • 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.

Complete orthogonal decomposition edit

  • 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.

Other decompositions edit

Polar decomposition edit

  • 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 edit

  • 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 edit

  • 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 edit

  • 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 edit

  • 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]

Williamson's normal form edit

  • Applicable to: square, positive-definite real matrix A with order 2n×2n.
  • Decomposition:  , where   is a symplectic matrix and D is a nonnegative n-by-n diagonal matrix.[12]

Matrix square root edit

  • Decomposition:  , not unique in general.
  • In the case of positive semidefinite  , there is a unique positive semidefinite   such that  .

Generalizations edit

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).

See also edit

References edit

Notes edit

  1. ^ If a non-square matrix is used, however, then the matrix U will also have the same rectangular shape as the original matrix A. And so, calling the matrix U upper triangular would be incorrect as the correct term would be that U is the 'row echelon form' of A. Other than this, there are no differences in LU factorization for square and non-square matrices.

Citations edit

  1. ^ Lay, David C. (2016). Linear algebra and its applications. Steven R. Lay, Judith McDonald (Fifth Global ed.). Harlow. p. 142. ISBN 978-1-292-09223-2. OCLC 920463015.{{cite book}}: CS1 maint: location missing publisher (link)
  2. ^ Piziak, R.; Odell, P. L. (1 June 1999). "Full Rank Factorization of Matrices". Mathematics Magazine. 72 (3): 193. doi:10.2307/2690882. JSTOR 2690882.
  3. ^ Uhlmann, J.K. (2018), "A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations", SIAM Journal on Matrix Analysis and Applications, 239 (2): 781–800, doi:10.1137/17M113890X
  4. ^ Uhlmann, J.K. (2018), "A Rank-Preserving Generalized Matrix Inverse for Consistency with Respect to Similarity", IEEE Control Systems Letters, 3: 91–95, arXiv:1804.07334, doi:10.1109/LCSYS.2018.2854240, ISSN 2475-1456, S2CID 5031440
  5. ^ Choudhury & Horn 1987, pp. 219–225
  6. ^ a b c 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., vol. 14, American Mathematical Society, pp. 31–54
  9. ^ Nielsen, Frank; Bhatia, Rajendra (2012). Matrix Information Geometry. Springer. p. 224. arXiv:1007.4402. doi:10.1007/978-3-642-30232-9. ISBN 9783642302329. S2CID 118466496.
  10. ^ Zhang, Fuzhen (30 June 2014). "A matrix decomposition and its applications". Linear and Multilinear Algebra. 63 (10): 2033–2042. doi:10.1080/03081087.2014.933219. S2CID 19437967.
  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. S2CID 119578994.
  13. ^ Townsend & Trefethen 2015

Bibliography edit

  • Choudhury, Dipa; Horn, Roger A. (April 1987). "A Complex Orthogonal-Symmetric Analog of the Polar Decomposition". SIAM Journal on Algebraic and Discrete Methods. 8 (2): 219–225. 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: 43–92. 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 (4): 433–476, doi:10.1007/bf01449770
  • Simon, C.; Blume, L. (1994). Mathematics for Economists. Norton. ISBN 978-0-393-95733-4.
  • 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): 20140585, Bibcode:2014RSPSA.47140585T, doi:10.1098/rspa.2014.0585, PMC 4277194, PMID 25568618
  • Jun, Lu (2021), Numerical matrix decomposition and its modern applications: A rigorous first course, arXiv:2107.02579

External links edit

  • Online Matrix Calculator
  • Wolfram Alpha Matrix Decomposition Computation » LU and QR Decomposition
  • Springer Encyclopaedia of Mathematics » Matrix factorization
  • GraphLab GraphLab collaborative filtering library, large scale parallel implementation of matrix decomposition methods (in C++) for multicore.