# Matrix decomposition

﻿
Matrix decomposition

In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. 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 − 1b 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 a unitary 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

### Cholesky decomposition

• Applicable to: square, symmetric, positive definite matrix A
• Decomposition: A = UTU, where U is upper triangular with positive diagonal entries
• Comment: the Cholesky decomposition is a special case of the symmetric LU decomposition, with L = UT.
• Comment: the Cholesky decomposition is unique
• Comment: the Cholesky decomposition is also applicable for complex hermitian positive definite matrices
• Comment: An alternative is the LDL decomposition which can avoid extracting square roots.

### QR decomposition

• Applicable to: m-by-n matrix A
• Decomposition: A = QR where Q is an orthogonal matrix of size m-by-m, and R is an upper triangular matrix of size m-by-n
• Comment: The QR decomposition provides an alternative way of solving the system of equations Ax = b without inverting the matrix A. The fact that Q is orthogonal means that QTQ = I, so that Ax = b is equivalent to Rx = QTb, which is easier to solve since R is triangular.

### Singular value decomposition

• Applicable to: m-by-n matrix A.
• Decomposition: A = UDVH, where D is a nonnegative diagonal matrix, and U and V are unitary matrices, and VH denotes 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 below, 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.

## Decompositions based on eigenvalues and related concepts

### Eigendecomposition

• Also called spectral decomposition
• Applicable to: square matrix A.
• Decomposition: A = VDV − 1, 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 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 AV = VD. If the n eigenvalues are distinct (that is, none is equal to any of the others), then V is invertible, implying the decomposition A = VDV − 1.
• 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 xt + 1 = Axt starting from the initial condition x0 = c is solved by xt = Atc, which is equivalent to xt = VDtV − 1c, where V and D are the matrices formed from the eigenvectors and eigenvalues of A. Since D is diagonal, raising it to power Dt, just involves raising each element on the diagonal to the power t. This is much easier to do and to 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.

### Schur decomposition

• Applicable to: square matrix A
• Comment: there are two versions of this decomposition: the complex Schur decomposition and the real Schur decomposition. A complex matrix always has a complex Schur decomposition. A real matrix admits a real Schur decomposition if and only if all of its eigenvalues are real.
• Decomposition (complex version): A = UTUH, where U is a unitary matrix, UH is the conjugate transpose of U, and T is an upper triangular matrix called the complex Schur form which has the eigenvalues of A along its diagonal.
• Decomposition (real version): A = VSVT, where A, V, S and VT are matrices that contain real numbers only. In this case, V is an orthogonal matrix, VT 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): A = QSZH and B = QTZH 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, λi = Sii / Tii, are the generalized eigenvalues that solve the generalized eigenvalue problem Av = λBv (where λ is an unknown scalar and v is an unknown nonzero vector).
• Decomposition (real version): A = QSZT and B = QTZT 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: A = VDVT, where D is a real nonnegative diagonal matrix, and V is unitary. VT denotes the matrix transpose of V.
• Comment: the diagonal elements of D are the nonnegative square roots of the eigenvalues of AAH.
• Comment: V may be complex even if A is real.

## Other decompositions

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Matrix - получить на Академике рабочий купон на скидку Летуаль или выгодно matrix купить с бесплатной доставкой на распродаже в Летуаль

• matrix decomposition — noun A process by which a rectangular table of numbers or abstract quantities that can be added and multiplied is broken down into simpler numerical building blocks …   Wiktionary

• Crout matrix decomposition — In linear algebra, the Crout matrix decomposition is an LU decomposition which decomposes a matrix into a lower triangular matrix (L), an upper triangular matrix (U) and, although not always needed, a permutation matrix (P). The Crout matrix… …   Wikipedia

• Decomposition (disambiguation) — Decomposition may refer to the following: Decomposition, biological process through which organic material is reduced Chemical decomposition or analysis, in chemistry, is the fragmentation of a chemical compound into elements or smaller compounds …   Wikipedia

• Matrix (mathematics) — Specific elements of a matrix are often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A. In mathematics, a matrix (plural matrices, or less commonly matrixes)… …   Wikipedia

• Decomposition en valeurs singulieres — Décomposition en valeurs singulières En mathématiques, le procédé d algèbre linéaire de décomposition en valeurs singulières (ou SVD, de l anglais : Singular Value Decomposition) d une matrice est un outil important de factorisation des… …   Wikipédia en Français

• Décomposition En Valeurs Singulières — En mathématiques, le procédé d algèbre linéaire de décomposition en valeurs singulières (ou SVD, de l anglais : Singular Value Decomposition) d une matrice est un outil important de factorisation des matrices rectangulaires réelles ou… …   Wikipédia en Français

• Matrix theory — is a branch of mathematics which focuses on the study of matrices. Initially a sub branch of linear algebra, it has grown to cover subjects related to graph theory, algebra, combinatorics, and statistics as well.HistoryThe term matrix was first… …   Wikipedia

• Décomposition en valeurs singulières — En mathématiques, le procédé d algèbre linéaire de décomposition en valeurs singulières (ou SVD, de l anglais : Singular Value Decomposition) d une matrice est un outil important de factorisation des matrices rectangulaires réelles ou… …   Wikipédia en Français

• Décomposition QR — En algèbre linéaire, la décomposition QR (appelée aussi, décomposition QU) d une matrice A est une décomposition de la forme A = QR où Q est une matrice orthogonale (QQT = I), et R une matrice triangulaire supérieure. Ce type de décomposition est …   Wikipédia en Français

• Matrix function — In mathematics, a matrix function is a function which maps a matrix to another matrix. Contents 1 Extending scalar functions to matrix functions 1.1 Power series 1.2 Jordan decomposition …   Wikipedia