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Representation of a matrix as a product
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
Matrix_decomposition
Matrix decomposition method
Cholesky decomposition or Cholesky factorization (pronounced /ʃəˈlɛski/ shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the
Cholesky_decomposition
Type of matrix factorization
lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix multiplication
LU_decomposition
mathematics, and in particular modular representation theory, a decomposition matrix is a matrix that results from writing the irreducible ordinary characters
Decomposition_matrix
Matrix decomposition
this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
Matrix decomposition
decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal matrix Q
QR_decomposition
Matrix factorisation in mathematics
decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as
Schur_decomposition
Matrix decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix into a rotation, followed by a scaling, followed
Singular_value_decomposition
Type of matrix factorization
the Crout matrix decomposition is an LU decomposition which decomposes a matrix into a lower triangular matrix (L), an upper triangular matrix (U) and,
Crout_matrix_decomposition
Type of matrix representation
In mathematics, the polar decomposition of a square real or complex matrix A {\displaystyle A} is a factorization of the form A = U P {\displaystyle A=UP}
Polar_decomposition
Process in algebra
The main tensor decompositions are: Tensor rank decomposition; Higher-order singular value decomposition; Tucker decomposition; matrix product states,
Tensor_decomposition
Special kind of square matrix
the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix L and an upper triangular matrix U if and
Triangular_matrix
Studies linear representations of finite groups over fields of positive characteristic
irreducible Brauer characters assigned columns. This is referred to as the decomposition matrix, and is frequently labelled D. It is customary to place the trivial
Modular_representation_theory
Topics referred to by the same term
Spectral decomposition is any of several things: Spectral decomposition for matrix: eigendecomposition of a matrix Spectral decomposition for linear operator:
Spectral_decomposition
Matrix with a multiplicative inverse
exist, it is necessary for the matrix involved to be invertible. Matrix decomposition techniques like LU decomposition are much faster than inversion
Invertible_matrix
Result about when a matrix can be diagonalized
{\displaystyle A} . When the matrix being decomposed is Hermitian, the spectral decomposition is a special case of the Schur decomposition (see the proof in case
Spectral_theorem
Property of a mathematical matrix
{\displaystyle M^{\frac {1}{2}}} for any such decomposition, or specifically for the Cholesky decomposition, or any decomposition of the form M = B B ; {\displaystyle
Definite_matrix
Kind of square matrix in linear algebra
Hessenberg matrix has zero entries above the first superdiagonal. They are named after Karl Hessenberg. A Hessenberg decomposition is a matrix decomposition of
Hessenberg_matrix
Most widely known generalized inverse of a matrix
pseudoinverse can be expressed using the singular value decomposition. Any matrix can be decomposed as A = U D V ∗ {\displaystyle A=UDV^{*}} for some isometries
Moore–Penrose_inverse
Decomposition in multilinear algebra
decomposition or rank-R decomposition is the decomposition of a tensor as a sum of R rank-1 tensors, where R is minimal. Computing this decomposition
Tensor_rank_decomposition
Matrix equal to its transpose
as sum of a symmetric and a skew-symmetric matrix. This decomposition is known as the Toeplitz decomposition. Let Mat n {\displaystyle {\mbox{Mat}}_{n}}
Symmetric_matrix
Dimensionality reduction algorithm
Eigenvalue decomposition Empirical mode decomposition Global mode Normal mode Proper orthogonal decomposition Singular-value decomposition Schmid, Peter
Dynamic_mode_decomposition
Norm on a vector space of matrices
singular value decomposition is useful in analyzing matrices. A vector norm of the singular values of a matrix may be taken as a matrix norm. Such norms
Matrix_norm
Topics referred to by the same term
mathematics, Jordan decomposition may refer to Hahn decomposition theorem, and the Jordan decomposition of a measure Jordan normal form of a matrix Jordan–Chevalley
Jordan_decomposition
Real square matrix whose columns and rows are orthogonal unit vectors
Singular value decomposition M = UΣVT, U and V orthogonal, Σ diagonal matrix Eigendecomposition of a symmetric matrix (decomposition according to the
Orthogonal_matrix
Mathematical process dealing with Lie groups
mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a
Iwasawa_decomposition
Algorithms for matrix decomposition
Nonnegative Matrix Factorization (DNMF), Scalable Nonnegative Matrix Factorization (ScalableNMF), Distributed Stochastic Singular Value Decomposition. Online:
Non-negative matrix factorization
Non-negative_matrix_factorization
in the decomposed matrix are essentially the same as their meanings in the original matrix. Formally, a CUR matrix approximation of a matrix A is three
CUR_matrix_approximation
Matrix B such that B² equals a given matrix A
distinct meaning is discussed in the article Definite matrix § Decomposition. In general, a matrix can have several square roots. In particular, if A = B 2
Square_root_of_a_matrix
Matrix that commutes with its conjugate transpose
The left and right singular vectors in the singular value decomposition of a normal matrix A = U D V ∗ {\displaystyle A=UDV^{*}} differ only in complex
Normal_matrix
Quantum state of multiple particles represented as complex matrices
as an MPS: left-canonical decomposition, right-canonical decomposition, and mixed-canonical decomposition. The decomposition of the d N {\displaystyle
Matrix_product_state
Array of numbers
easier. The LU decomposition factors matrices as a product of lower (L) and an upper triangular matrices (U). Once this decomposition is calculated, linear
Matrix_(mathematics)
Mathematical concept
This decomposition is closely related to the singular value decomposition of a matrix and is known as an 'Euler' or 'Bloch-Messiah' decomposition. The
Symplectic_matrix
Matrix equal to its conjugate-transpose
matrices also appear in techniques like singular value decomposition (SVD) and eigenvalue decomposition. In statistics and machine learning, Hermitian matrices
Hermitian_matrix
Type of matrix factorization
Block LU decomposition is a matrix decomposition of a block matrix into a lower block triangular matrix L and an upper block triangular matrix U. This
Block_LU_decomposition
Complex matrix whose conjugate transpose equals its inverse
factorizations of a unitary matrix in basic matrices are possible. Hermitian matrix Skew-Hermitian matrix Matrix decomposition Orthogonal group O(n) Special
Unitary_matrix
matrix Hankel matrix (0,1)-matrix Bohemian matrices Matrix decomposition Cholesky decomposition LU decomposition QR decomposition Polar decomposition
Outline_of_linear_algebra
Dimension of the column space of a matrix
computers, basic Gaussian elimination (LU decomposition) can be unreliable, and a rank-revealing decomposition should be used instead. An effective alternative
Rank_(linear_algebra)
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
Frobenius normal form Jordan matrix Jordan–Chevalley decomposition Matrix decomposition Modal matrix Weyr canonical form Shilov defines the term Jordan
Jordan_normal_form
Matrix with shifting rows
{\displaystyle O(n^{2})} time. The Bareiss algorithm for an LU decomposition is stable. An LU decomposition gives a quick method for solving a Toeplitz system, and
Toeplitz_matrix
Method of data analysis
multivariate quality control, proper orthogonal decomposition (POD) in mechanical engineering, singular value decomposition (SVD) of X (invented in the last quarter
Principal_component_analysis
Generalization of gamma distribution to multiple dimensions
covariance matrix of a multivariate normal distribution. A derivation of the MLE uses the spectral theorem. The Bartlett decomposition of a matrix X from
Wishart_distribution
Square matrix without an inverse
A singular matrix is a square matrix that is not invertible, unlike non-singular matrices which are invertible. Equivalently, an n {\displaystyle n} -by-
Singular_matrix
Matrix representing a Euclidean rotation
rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix R = [
Rotation_matrix
algebra, the complete orthogonal decomposition is a matrix decomposition. It is similar to the singular value decomposition, but typically somewhat cheaper
Complete orthogonal decomposition
Complete_orthogonal_decomposition
Concept in linear algebra
a matrix decomposition algorithm based on the QR factorization which can be used to determine the rank of a matrix. The singular value decomposition can
RRQR_factorization
Mathematical term
maximal torus of B {\displaystyle B} . The Bruhat decomposition of G {\displaystyle G} is the decomposition G = B W B = ⨆ w ∈ W B w B {\displaystyle G=BWB=\bigsqcup
Bruhat_decomposition
Process in linear algebra
unique up to re-ordering. The Schmidt decomposition is essentially a restatement of the singular value decomposition in a different context. Fix orthonormal
Schmidt_decomposition
Orthonormalization of a set of vectors
vectors of a full column rank matrix yields the QR decomposition (it is decomposed into an orthogonal and a triangular matrix). The vector projection of
Gram–Schmidt_process
Matrices named after Élie Cartan
the above decomposition is positive definite, then A is said to be a Cartan matrix. The Cartan matrix of a simple Lie algebra is the matrix whose elements
Cartan_matrix
Square matrix in which each ascending skew-diagonal from left to right is constant
decomposition of the Hankel matrix provides a means of computing the A, B, and C matrices which define the state-space realization. The Hankel matrix
Hankel_matrix
In mathematics, invariant of square matrices
decomposition methods. Examples include the LU decomposition, the QR decomposition or the Cholesky decomposition (for positive definite matrices). These methods
Determinant
Topics referred to by the same term
manifolds JSJ decomposition, or toral decomposition, a decomposition of 3-manifolds Matrix decomposition, a factorization of a matrix into a product
Decomposition (disambiguation)
Decomposition_(disambiguation)
Matrix in which most of the elements are zero
the matrix. The symbolic Cholesky decomposition can be used to calculate the worst possible fill-in before doing the actual Cholesky decomposition. There
Sparse_matrix
Matrix representation of a graph
— as established by Cheeger's inequality. The spectral decomposition of the Laplacian matrix allows the construction of low-dimensional embeddings that
Laplacian_matrix
Polynomial Matrix Spectral Factorization or Matrix Fejér–Riesz Theorem is a tool used to study the matrix decomposition of polynomial matrices. Polynomial
Polynomial matrix spectral factorization
Polynomial_matrix_spectral_factorization
Generalized matrix decomposition for Lie groups and Lie algebras
In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure
Cartan_decomposition
Field of mathematics
problems is a reason to favour matrix decomposition methods like using the singular value decomposition. Some matrix decomposition methods may be unstable,
Numerical_linear_algebra
Locality condition in quantum field theory
changing the S-matrix, which would violate cluster decomposition. This means that in momentum space cluster decomposition requires that the S-matrix only has
Cluster_decomposition
reducing a general matrix into a bidiagonal one, and the singular value decomposition (SVD) uses this method as well. Reducing a matrix to bidiagonal form
Bidiagonal_matrix
Matrix of inner products of vectors
The diagonalization of the Gram matrix is the singular value decomposition. The Gram matrix is symmetric in the case the inner product is real-valued; it
Gram_matrix
Jordan–Chevalley decomposition of an element in algebraic group as a product of semisimple and unipotent elements The Bruhat decomposition G = B W B {\displaystyle
Lie_group_decomposition
Algorithm to calculate eigenvalues
idea is to perform a QR decomposition, writing the matrix as a product of an orthogonal matrix and an upper triangular matrix, multiply the factors in
QR_algorithm
Method of data analysis
PCA, which aims to recover a low-rank matrix L0 from highly corrupted measurements M = L0 +S0. This decomposition in low-rank and sparse matrices can be
Robust principal component analysis
Robust_principal_component_analysis
Measure of covariance of components of a random vector
covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the
Covariance_matrix
Signal processing algorithm
Empirical Mode Decomposition have been used to analyze characterization of multidimensional signals. The empirical mode decomposition (EMD) method can
Multidimensional empirical mode decomposition
Multidimensional_empirical_mode_decomposition
interpolative decomposition (ID) factors a matrix as the product of two matrices, one of which contains selected columns from the original matrix, and the
Interpolative_decomposition
Mathematical expression for linear operators
Jordan–Chevalley decomposition also exist for elements of Linear algebraic groups and Lie groups via a multiplicative reformulation. The decomposition is an important
Jordan–Chevalley decomposition
Jordan–Chevalley_decomposition
Concepts from linear algebra
(PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix is used
Eigenvalues_and_eigenvectors
Algorithmic runtime requirements for matrix multiplication
true LU decomposition of the original matrix. The argument applies also for the determinant, since it results from the block LU decomposition that det
Computational complexity of matrix multiplication
Computational_complexity_of_matrix_multiplication
SimpleMatrix result = matA.mult(matB); Use of a DecompositionFactory to compute a Singular Value Decomposition with a Dense Double Row Major matrix (DDRM):
Efficient_Java_Matrix_Library
Numerical method that reduces the complexity of computationally intensive simulations
Proper Orthogonal Decomposition (POD), as it was originally formulated in the domain of fluid dynamics to analyze turbulences, is to decompose a random vector
Proper orthogonal decomposition
Proper_orthogonal_decomposition
For a square matrix, the transpose of the cofactor matrix
classical adjoint adj(A) of a square matrix A is the transpose of its cofactor matrix. It is occasionally known as adjunct matrix, or "adjoint", though that normally
Adjugate_matrix
Topics referred to by the same term
presentation of an invertible matrix with polynomial coefficients as a product of three matrices. The Birkhoff - von Neumann decomposition, introduced by Garrett
Birkhoff_decomposition
Name of two different techniques based on the singular value decomposition
generalized singular value decomposition (GSVD) is the name of two different techniques based on the singular value decomposition (SVD). The two versions
Generalized singular value decomposition
Generalized_singular_value_decomposition
Certain vector fields are the sum of an irrotational and a solenoidal vector field
rotation field. This decomposition may be calculated for vector fields that satisfy certain regularity or decay conditions. A decomposition exists for all smooth
Helmholtz_decomposition
Mapping of a graph into a tree
constraint satisfaction, query optimization, and matrix decomposition. The concept of tree decomposition was originally introduced by Rudolf Halin (1976)
Tree_decomposition
Algorithmic technique using hashing
indexing Rolling hash – Type of hash function Singular value decomposition – Matrix decomposition Sparse distributed memory – Mathematical model of memory
Locality-sensitive_hashing
Technique in mathematical modeling
into this class but are perhaps less common are: Proper generalized decomposition Matrix interpolation Transfer function interpolation Piecewise tangential
Model_order_reduction
Matrix of second derivatives
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function
Hessian_matrix
Topics referred to by the same term
primality test for Mersenne numbers Cholesky decomposition, an algorithm to decompose matrix A into a lower Matrix L : A = LLT. Linus Media Group, a tech media
LLT
Matrix of partial derivatives of a vector-valued function
vector calculus, the Jacobian matrix (/dʒəˈkoʊbiən/, /dʒɪ-, jɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Matrix whose only nonzero elements are on its main diagonal
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices
Diagonal_matrix
Matrix of geometric progressions
In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row:
Vandermonde_matrix
Dictionary learning algorithm
E_{k}} term with a r a n k − 1 {\displaystyle rank-1} matrix using singular value decomposition, then update d k {\displaystyle d_{k}} with it. However
K-SVD
Matrices similar to diagonal matrices
defective matrix can be deformed into a diagonalizable matrix by a small perturbation; and the Jordan–Chevalley decomposition states that any matrix is uniquely
Diagonalizable_matrix
Equation in economics
the Slutsky equation. This process is sometimes known as the Hicks decomposition of a demand change. The equation can be rewritten in terms of elasticity:
Slutsky_equation
Theorem of matrix ranks
inverse of the matrix A + B where the matrix B can be approximated by a low-rank matrix UCV, for example using the singular value decomposition. This is applied
Woodbury_matrix_identity
Mathematical procedure
recommender systems. Matrix factorization algorithms work by decomposing the user-item interaction matrix into the product of two lower dimensionality rectangular
Matrix factorization (recommender systems)
Matrix_factorization_(recommender_systems)
Approximation of a matrix's Cholesky factorization
applying Cholesky decomposition, we observe that some zero elements in the original matrix end up being non-zero elements in the decomposed matrix, like elements
Incomplete Cholesky factorization
Incomplete_Cholesky_factorization
Tensor decomposition
value decomposition (HOSVD) is a misnomer. There does not exist a single tensor decomposition that retains all the defining properties of the matrix SVD
Higher-order singular value decomposition
Higher-order_singular_value_decomposition
Optical phenomenon
coherence of a spectral decomposition of the signal, while the Wolf coherency matrix averages over all time/frequencies. The coherency matrix contains all second
Unpolarized_light
Markov chains with this property. Ando and Fisher define a completely decomposable matrix as one where "an identical rearrangement of rows and columns leaves
Nearly completely decomposable Markov chain
Nearly_completely_decomposable_Markov_chain
Concept in linear algebra
multiplication of unitary matrices is itself a unitary matrix, this gives us the unitary matrix of the QR decomposition). Let e 1 , e 2 , ... , e n {\displaystyle
Householder_transformation
Set of a matrix's eigenvalues
spectral decomposition) of a diagonalizable matrix is a decomposition of a diagonalizable matrix into a specific canonical form whereby the matrix is represented
Spectrum_of_a_matrix
Polynomial whose roots are the eigenvalues of a matrix
repeated. Moreover, the Jordan decomposition theorem guarantees that any square matrix A {\displaystyle A} can be decomposed as A = S − 1 U S , {\displaystyle
Characteristic_polynomial
{\beta }}}.} The matrix X is subjected to an orthogonal decomposition, e.g., the QR decomposition as follows. X = Q ( R 0 ) {\displaystyle
Numerical methods for linear least squares
Numerical_methods_for_linear_least_squares
Canonical form of matrices over a field
must instead use a decomposition into generalized eigenspaces, and possibly into Jordan blocks. But obtaining such a fine decomposition is not necessary
Frobenius_normal_form
Matrix which differs from the identity matrix by one elementary row operation
mathematics, an elementary matrix is a square matrix obtained from the application of a single elementary row operation to the identity matrix. The elementary matrices
Elementary_matrix
Topics referred to by the same term
pounds avoirdupois QR decomposition, a decomposition of a matrix QR algorithm, an eigenvalue algorithm to perform QR decomposition Quadratic reciprocity
QR
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DECOMPOSITION MATRIX
DECOMPOSITION MATRIX
DECOMPOSITION MATRIX
DECOMPOSITION MATRIX
DECOMPOSITION MATRIX
DECOMPOSITION MATRIX
DECOMPOSITION MATRIX
DECOMPOSITION MATRIX
DECOMPOSITION MATRIX
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