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Algebra, a branch of mathematics
In homological algebra, a branch of mathematics, a matrix factorization is a tool used to study infinitely long resolutions, generally over commutative
Matrix factorization (algebra)
Matrix_factorization_(algebra)
Representation of a matrix as a product
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;
Matrix_decomposition
(Mathematical) decomposition into a product
example, 3 × 5 is an integer factorization of 15, and (x − 2)(x + 2) is a polynomial factorization of x2 − 4. Factorization is not usually considered meaningful
Factorization
Algorithms for matrix decomposition
Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where
Non-negative matrix factorization
Non-negative_matrix_factorization
Field of mathematics
numerical linear algebra include obtaining matrix decompositions like the singular value decomposition, the QR factorization, the LU factorization, or the eigendecomposition
Numerical_linear_algebra
Matrix decomposition
In linear algebra, eigendecomposition (also known as eigenvalue decomposition or EVD) is a factorization of a matrix A {\displaystyle A} into a canonical
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
Computational method
In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field
Factorization_of_polynomials
Complex matrix whose conjugate transpose equals its inverse
In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U−1 equals its conjugate transpose U*, that is, if U ∗ U = U
Unitary_matrix
as Positivstellensatz. Likewise, the Polynomial Matrix Spectral Factorization provides a factorization for positive definite polynomial matrices. This
Polynomial matrix spectral factorization
Polynomial_matrix_spectral_factorization
Dimension of the column space of a matrix
In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This corresponds to the maximal
Rank_(linear_algebra)
Type of matrix factorization
algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see
LU_decomposition
Matrix decomposition
In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of
QR_decomposition
Algorithm to multiply matrices
Demmel, James (2011). "Communication-optimal parallel 2.5D matrix multiplication and LU factorization algorithms" (PDF). Proceedings of the 17th International
Matrix multiplication algorithm
Matrix_multiplication_algorithm
Matrix decomposition method
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced /ʃəˈlɛski/ shə-LES-kee) is a decomposition of a Hermitian, positive-definite
Cholesky_decomposition
Approximation of a matrix's Cholesky factorization
factorization of a symmetric positive definite matrix is a sparse approximation of the Cholesky factorization. An incomplete Cholesky factorization is
Incomplete Cholesky factorization
Incomplete_Cholesky_factorization
Matrix defined using smaller matrices called blocks
y\in {\text{colgroups}}} . Block matrix algebra arises in general from biproducts in categories of matrices. The matrix P = [ 1 2 2 7 1 5 6 2 3 3 4 5 3
Block_matrix
Concept in numerical linear algebra
numerical linear algebra, an incomplete LU factorization (abbreviated as ILU) of a matrix is a sparse approximation of the LU factorization often used as
Incomplete_LU_factorization
In linear algebra, the inverse square matrix A {\displaystyle A} is another square matrix A − 1 {\displaystyle A^{-1}} such that the product A − 1 A {\displaystyle
Methods_of_matrix_inversion
In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition
Factorization of polynomials over finite fields
Factorization_of_polynomials_over_finite_fields
algebra Factorization algebra Genetic algebra Geometric algebra Gerstenhaber algebra Graded algebra Griess algebra Group algebra Group algebra of a locally
List_of_algebras
Accomplishments in factoring large integers
Integer factorization is the process of determining which prime numbers divide a given positive integer. Doing this quickly has applications in cryptography
Integer_factorization_records
Concept in linear algebra
and a matrix A ∈ F m × n {\displaystyle A\in \mathbb {F} ^{m\times n}} , a rank decomposition or rank factorization of A is a factorization of A of
Rank_factorization
Finite extension of the rationals
study of rings of algebraic integers. For general Dedekind rings, in particular rings of integers, there is a unique factorization of ideals into a product
Algebraic_number_field
Array of numbers
infinite matrix. In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an empty matrix. The
Matrix_(mathematics)
Branch of number theory
arithmetic, that every (positive) integer has a factorization into a product of prime numbers, and this factorization is unique up to the ordering of the factors
Algebraic_number_theory
Kind of square matrix in linear algebra
linear algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular. To be exact, an upper Hessenberg matrix has zero
Hessenberg_matrix
Algebraic structure with addition and multiplication
⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields A ring is a set
Ring_(mathematics)
Scientific area at the interface between computer science and mathematics
differentiation using the chain rule, polynomial factorization, indefinite integration, etc. Computer algebra is widely used to experiment in mathematics and
Computer_algebra
In mathematics, invariant of square matrices
Nicolas (1998), Algebra I, Chapters 1-3, Springer, ISBN 9783540642435 Bunch, James R.; Hopcroft, John E. (1974). "Triangular Factorization and Inversion
Determinant
Polynomial whose roots are the eigenvalues of a matrix
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues
Characteristic_polynomial
Computer system for solving algebra problems
polynomials. Integer factorization algorithms include the Elliptic Curve Method, the Quadratic sieve and the Number field sieve. Algebraic number theory Magma
Magma (computer algebra system)
Magma_(computer_algebra_system)
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
Mathematical operation
square root may be used for any factorization of a positive semidefinite matrix A as BTB = A, as in the Cholesky factorization, even if BB ≠ A. This distinct
Square_root_of_a_matrix
Branch of mathematics
development, such as Boolean algebra, vector algebra, and matrix algebra. Influential early developments in abstract algebra were made by the German mathematicians
Algebra
Mathematical software
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in
Computer_algebra_system
Algebraic structure
factorization, as there are factorization algorithms that have a polynomial complexity. They are implemented in most general purpose computer algebra
Polynomial_ring
Matrix representing a Euclidean rotation
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention
Rotation_matrix
Matrix equal to its transpose
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, A is symmetric ⟺ A = A T . {\displaystyle A{\text{
Symmetric_matrix
Result about when a matrix can be diagonalized
In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented
Spectral_theorem
variable. column vector A matrix with only one column. complex number An element of a complex plane complex plane A linear algebra over the real numbers with
Glossary_of_linear_algebra
Exact sequence used to describe the structure of an object
Hilbert–Burch theorem Hilbert's syzygy theorem Free presentation Matrix factorizations (algebra) Jacobson 2009, §6.5 uses coresolution, though right resolution
Resolution_(algebra)
Method in computational algebra
Berlekamp, Elwyn R. (1968). Algebraic Coding Theory. McGraw Hill. ISBN 0-89412-063-8. Knuth, Donald E (1997). "4.6.2 Factorization of Polynomials". Seminumerical
Berlekamp's_algorithm
Type of mathematical expression
form, called factorization is, in general, too difficult to be done by hand-written computation. However, efficient polynomial factorization algorithms
Polynomial
Norm on a vector space of matrices
Applied Numerical Linear Algebra, section 1.7, published by SIAM, 1997. Carl D. Meyer, Matrix Analysis and Applied Linear Algebra, published by SIAM, 2000
Matrix_norm
Every polynomial has a real or complex root
proof: https://mizar.org/version/current/html/polynom5.html#T74 Prime Factorization Method — Prime Factorization Method explained in detail with Example.
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Matrix with shifting rows
In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to
Toeplitz_matrix
Mathematical technique
In mathematics, a matrix factorization of a polynomial is a technique for factoring irreducible polynomials with matrices. David Eisenbud proved that every
Matrix factorization of a polynomial
Matrix_factorization_of_a_polynomial
Concept in linear algebra
An RRQR factorization or rank-revealing QR factorization is a matrix decomposition algorithm based on the QR factorization which can be used to determine
RRQR_factorization
Algebraic structure used in analysis
In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket
Lie_algebra
Greatest common divisor of polynomials
computations provide the complete square-free factorization of the polynomial, which is a factorization f = ∏ i = 1 deg ( f ) f i i {\displaystyle f=\prod
Polynomial greatest common divisor
Polynomial_greatest_common_divisor
Nonlinear equation which arises on linear optimal control problems
of the complex plane. A solution to the algebraic Riccati equation can be obtained by matrix factorizations or by iterating on the Riccati equation.
Algebraic_Riccati_equation
Mathematical structure in abstract algebra
mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra; read as "star-algebra") is a mathematical structure consisting of
*-algebra
1966 mathematics textbook by Serge Lang
introduces the polynomial ideal as an algebraic structure, proving basic results about division and factorization before applying ideals in the decomposition
Linear_Algebra_(book)
Canonical form of matrices over a field
In linear algebra, the Frobenius normal form or rational canonical form of a square matrix A with entries in a field F is a canonical form for matrices
Frobenius_normal_form
Matrix in which most of the elements are zero
support for several sparse matrix formats, linear algebra, and solvers. ALGLIB is a C++ and C# library with sparse linear algebra support ARPACK Fortran 77
Sparse_matrix
Square matrix in which each ascending skew-diagonal from left to right is constant
In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a rectangular matrix in which each ascending skew-diagonal
Hankel_matrix
Software library for numerical linear algebra
decomposition. It also includes routines to implement the associated matrix factorizations such as LU, QR, Cholesky and Schur decomposition. The routines handle
LAPACK
comparison of linear algebra software libraries, either specialized or general purpose libraries with significant linear algebra coverage. Matrix types (special
Comparison of linear algebra libraries
Comparison_of_linear_algebra_libraries
Central object in linear algebra; mapping vectors to vectors
Transformation geometry Gentle, James E. (2007). "Matrix Transformations and Factorizations". Matrix Algebra: Theory, Computations, and Applications in Statistics
Transformation_matrix
Algebraic structure with only one element
In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton
Zero_object_(algebra)
Branch of functional analysis
subspace lattice algebras, many limit algebras. Banach algebra – Particular kind of algebraic structure Matrix mechanics – Formulation of quantum mechanics
Operator_algebra
Concept in machine learning
In 2009, the work of Sutskever introduced Bayesian Clustered Tensor Factorization to model relational concepts while reducing the parameter space. From
Tensor_(machine_learning)
Matrix decomposition in mathematics
decomposition (i.e. Gauss elimination) to loop groups. The factorization of an invertible matrix M ∈ G L n ( C [ z , z − 1 ] ) {\displaystyle M\in \mathrm
Birkhoff_factorization
In graph theory, the Tutte matrix A of a graph G = (V, E) is a matrix used to determine the existence of a perfect matching: that is, a set of edges which
Tutte_matrix
Type of matrix representation
complex matrix A {\displaystyle A} is a factorization of the form A = U P {\displaystyle A=UP} , where U {\displaystyle U} is a unitary matrix, and P {\displaystyle
Polar_decomposition
Algebra based on a vector space with a quadratic form
mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure
Clifford_algebra
Method of data analysis
matrix whose columns are orthogonal unit vectors of length p and called the right singular vectors of X. In terms of this factorization, the matrix XTX
Principal_component_analysis
Linear algebra concept
In linear algebra, a semi-orthogonal matrix is a non-square matrix with real entries where: if the number of columns exceeds the number of rows, then the
Semi-orthogonal_matrix
Matrix that converges to zero matrix
linear algebra, a convergent matrix is a matrix that converges to the zero matrix under matrix exponentiation. When successive powers of a matrix T become
Convergent_matrix
Ring that is also a vector space or a module
over a commutative ring K, with the usual matrix multiplication. A commutative algebra is an associative algebra for which the multiplication is commutative
Associative_algebra
Matrix class
In mathematics, a Cauchy matrix, named after Augustin-Louis Cauchy, is an m×n matrix with elements aij in the form a i j = 1 x i − y j ; x i − y j ≠ 0
Cauchy_matrix
Associative algebra used in combinatorics
operations being ordinary matrix addition, scaling and multiplication. The multiplicative identity element of the incidence algebra is the delta function
Incidence_algebra
In linear algebra, the complete orthogonal decomposition is a matrix decomposition. It is similar to the singular value decomposition, but typically somewhat
Complete orthogonal decomposition
Complete_orthogonal_decomposition
Algebraic structure where all polynomials have roots
⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields As an example,
Algebraically_closed_field
Algorithmic runtime requirements for common math procedures
Bunch, James R.; Hopcroft, John E. (1974). "Triangular Factorization and Inversion by Fast Matrix Multiplication". Mathematics of Computation. 28 (125):
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Algorithm for computing greatest common divisors
essential step in several integer factorization algorithms, such as Pollard's rho algorithm, Shor's algorithm, Dixon's factorization method and the Lenstra elliptic
Euclidean_algorithm
Finding values for variables that make an equation true
more generally algebraic varieties or manifolds. In particular, algebraic geometry may be viewed as the study of solution sets of algebraic equations. The
Equation_solving
Area of discrete mathematics
linear algebra and group theory. A study of graph theory using linear algebra is called spectral graph theory. This study focuses on adjacency matrix, a matrix
Graph_theory
Quantum consistency equation
was the theory of factorized S-matrix in two dimensional quantum field theory. Alexander B. Zamolodchikov pointed out that the algebraic mechanics working
Yang–Baxter_equation
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
Approximation method
offer a major advantage: the results of matrix arithmetic operations like matrix multiplication, factorization or inversion can be approximated in O (
Hierarchical_matrix
Type of mathematical equation
derivatives of various orders. A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions to
Matrix_differential_equation
Group of mathematical theorems
groups is (normal epi, mono)-factorizable; in other words, the normal epimorphisms and the monomorphisms form a factorization system for the category. This
Isomorphism_theorems
Algorithm in number theory
theory, Dixon's factorization method (also Dixon's random squares method or Dixon's algorithm) is a general-purpose integer factorization algorithm; it
Dixon's_factorization_method
Computer algebra system
algebra system. It consists of an interpreter environment, a compiler and a library, which defines a strongly typed hierarchy. Two computer algebra systems
Axiom (computer algebra system)
Axiom_(computer_algebra_system)
Subclass of matrices
Gaussian elimination (LU factorization). The Jacobi and Gauss–Seidel methods for solving a linear system converge if the matrix is strictly (or irreducibly)
Diagonally_dominant_matrix
In linear algebra, relation between 3 dimensions
The rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity
Rank–nullity_theorem
Theorem
mathematics, Stinespring's dilation theorem, also called Stinespring's factorization theorem, named after W. Forrest Stinespring,[when?] is a result from
Stinespring_dilation_theorem
Matrix factorisation in mathematics
In linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary
Schur_decomposition
Mathematical study of linear operators
collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory. Single
Operator_theory
Class of commutative rings
algebraically independent set of generators of a field extension F. A seed consists of a cluster {x, y, ...} of F, together with an exchange matrix B
Cluster_algebra
Branch of algebra
unique factorization domain ⊂ integral domain ⊂ commutative ring. Algebraic geometry is in many ways the mirror image of commutative algebra. This correspondence
Ring_theory
Branch of mathematics that studies algebraic structures
algebra in Wiktionary, the free dictionary. In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures
List of abstract algebra topics
List_of_abstract_algebra_topics
Topics referred to by the same term
theorem of algebra, a theorem regarding the factorization of polynomials Fundamental theorem of arithmetic, a theorem regarding prime factorization Fundamental
Fundamental
Hypercomplex number system
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented
Octonion
R[x], then any factorization of its image P in (R/m)[x] into a product of coprime monic polynomials can be lifted to a factorization in R[x]. 2. A Henselian
Glossary of commutative algebra
Glossary_of_commutative_algebra
the polynomial and its derivative. The square-free factorization of a polynomial p is a factorization p = p 1 p 2 2 ⋯ p k k {\displaystyle p=p_{1}p_{2}^{2}\cdots
Polynomial_root-finding
Polynomial associated with a matrix
In linear algebra, the minimal polynomial μA of an n × n {\displaystyle n\times n} matrix A over a field F is the monic polynomial μA over F of least
Minimal polynomial (linear algebra)
Minimal_polynomial_(linear_algebra)
Algebra used in 2D conformal field theories and string theory
D-module-theoretic objects called chiral algebras introduced by Alexander Beilinson and Vladimir Drinfeld and factorization algebras, also introduced by Beilinson
Vertex_operator_algebra
In linear algebra, the nonnegative rank of a nonnegative matrix is a concept similar to the usual linear rank of a real matrix, but adding the requirement
Nonnegative rank (linear algebra)
Nonnegative_rank_(linear_algebra)
MATRIX FACTORIZATION-ALGEBRA
MATRIX FACTORIZATION-ALGEBRA
Male
English
Pet form of English Martin, MARTIE means "of/like Mars."
Male
French
French and German form of Greek Mattathias, MATHIS means "gift of God."
Female
German
Pet form of German Katarine, KATRIN means "pure."
Male
Hungarian
Czech and Hungarian form of Greek Patrikios, PATRIK means "patrician, of noble descent."
Female
Finnish
Pet form of Finnish Katariina, KATRI means "pure."
Male
English
 English form of Roman Latin Martinus, MARTIN means "of/like Mars." Compare with another form of Martin.
Male
French
 French form of Roman Latin Martinus, MARTIN means "of/like Mars." Compare with another form of Martin.
Female
Finnish
Finnish form of Greek Margarites, MAARIT means "pearl."
Girl/Female
Arabic, Australian, Basque, French, Latin
Lady; Feminine of Martin; Warlike
Surname or Lastname
English (of Welsh origin)
English (of Welsh origin) : variant of Maddox.
Girl/Female
Biblical
Rain, prison.
Female
English
English form of Latin Viatrix, BEATRIX means "voyager (through life)."
Female
Finnish
Finnish form of Greek Maria, MAARIA means "obstinacy, rebelliousness" or "their rebellion."Â
Female
English
Pet form of English Matilda, MATTIE means "mighty in battle." Compare with masculine Mattie.
Male
English
Pet form of English Matthew, MATTIE means "gift of God." Compare with feminine Mattie.
Female
Welsh
Welsh form of Old French Caterine, CATRIN means "pure."
Male
Italian
Italian form of Hebrew Mattithyah, MATTIA means "gift of God."
Female
English
French form of Latin Maria, MARIE means "obstinacy, rebelliousness" or "their rebellion."
Male
English
Anglicized form of Irish Gaelic MainchÃn, MANNIX means "little monk."
Girl/Female
Maori
The Maori form of April.
MATRIX FACTORIZATION-ALGEBRA
MATRIX FACTORIZATION-ALGEBRA
Boy/Male
Tamil
Manwendra | மாநவேநà¯à®¤à¯à®°Â Â
King among men
Boy/Male
Hindu, Indian
Lord
Boy/Male
American, Anglo, British, English
From the Stag's Ford
Surname or Lastname
English
English : variant of Keville.
Girl/Female
Bengali, Indian
Prince of Lord Shiva
Girl/Female
Indian, Telugu
Hope
Girl/Female
Tamil
New
Female
Hungarian
Hungarian form of Greek Hanna, ANIKÓ means "favor; grace."
Boy/Male
Biblical
Opening.
Female
Irish
Diminutive form of Irish Gaelic BrÃd, BRÃDIN means "little exalted one."
MATRIX FACTORIZATION-ALGEBRA
MATRIX FACTORIZATION-ALGEBRA
MATRIX FACTORIZATION-ALGEBRA
MATRIX FACTORIZATION-ALGEBRA
MATRIX FACTORIZATION-ALGEBRA
n.
The cavity in which anything is formed, and which gives it shape; a die; a mold, as for the face of a type.
n.
See Matrix.
a.
Of or pertaining to the meter as a standard of measurement; of or pertaining to the decimal system of measurement of which a meter is the unit; as, the metric system; a metric measurement.
v. t.
The white fibrous matter forming the matrix from which fungi.
n.
The martin.
n.
The womb.
n.
Hence, that which gives form or origin to anything
n.
A mold; a matrix.
n.
The five simple colors, black, white, blue, red, and yellow, of which all the rest are composed.
n.
A genus of swallows including the purple martin. See Martin.
n.
In type founding and forging, an impression or matrix, formed by a punch drift.
pl.
of Maori
n.
The lifeless portion of tissue, either animal or vegetable, situated between the cells; the intercellular substance.
v. i.
The mineral substance which incloses a vein; a matrix; a gangue.
pl.
of Matrix
n.
A housekeeper; esp., a woman who manages the domestic economy of a public instution; a head nurse in a hospital; as, the matron of a school or hospital.
n.
A rectangular arrangement of symbols in rows and columns. The symbols may express quantities or operations.
n.
The earthy or stony substance in which metallic ores or crystallized minerals are found; the gangue.
a.
Of or pertaining to the Maoris or to their language.