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111 relations: Band matrix, Bloch wave – MoM method, Block reflector, Bootstrapping (finance), Borel subgroup, Central series, Characteristic polynomial, Cholesky decomposition, Commuting matrices, Complexification (Lie group), Computational complexity of mathematical operations, Condition number, Cronbach's alpha, Crout matrix decomposition, Derivative of the exponential map, Determinant, Determinant identities, Diagonal matrix, Diagonalizable matrix, Divided differences, E8 lattice, Eigendecomposition of a matrix, Eigenvalue algorithm, Eigenvalues and eigenvectors, Elementary matrix, Engel's theorem, Factorization, Flag (linear algebra), Gauss–Seidel method, Gaussian elimination, Generalized flag variety, Generalized minimal residual method, Givens rotation, Glossary of Lie algebras, Gram–Schmidt process, Heisenberg group, Hessenberg matrix, Iliffe vector, Incomplete Cholesky factorization, Incomplete LU factorization, Irreducibility (mathematics), Iterative method, Iwasawa decomposition, Joel Lee Brenner, Jordan matrix, Jordan normal form, Kalman filter, Keller's conjecture, Kuiper's theorem, LAPACK, ..., Laplace expansion, Lattice (discrete subgroup), Lexicostatistics, Lie group decomposition, Lie–Kolchin theorem, Linear algebra, Linear algebraic group, Linear least squares (mathematics), List of linear algebra topics, List of matrices, List of numerical analysis topics, List of Runge–Kutta methods, List of terms relating to algorithms and data structures, List of triangle topics, LU decomposition, Matrix (mathematics), Matrix analysis, Matrix decomposition, Matrix group, Matrix ring, Matrix splitting, Metabelian group, Modular group, Molecular phylogenetics, Moore–Penrose inverse, Nest algebra, Newton polynomial, Nilpotent Lie algebra, Nilpotent matrix, Normal matrix, Packed storage matrix, Pascal matrix, Polyphase matrix, Positive-definite matrix, Projective variety, QR algorithm, QR decomposition, Regular chain, Riordan array, Schur decomposition, Separation principle, Serial module, Split Lie algebra, Square matrix, Stiff equation, Stirling number, Stirling numbers of the first kind, Stirling numbers of the second kind, Strict-feedback form, Successive over-relaxation, Sylvester equation, Sylvester's criterion, Trace (linear algebra), Triangular array, Triangular matrix, Triangular network coding, Triangulation (disambiguation), Vectorization (mathematics), Volodin space, Weight (representation theory), Zappa–Szép product. Expand index (61 more) »
Band matrix
In mathematics, particularly matrix theory, a band matrix is a sparse matrix whose non-zero entries are confined to a diagonal band, comprising the main diagonal and zero or more diagonals on either side.
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Bloch wave – MoM method
Bloch wave – MoM is a first principles technique for determining the photonic band structure of triply-periodic electromagnetic media such as photonic crystals.
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Block reflector
"A block reflector is an orthogonal, symmetric matrix that reverses a subspace whose dimension may be greater than one." It is built out of many elementary reflectors.
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Bootstrapping (finance)
In finance, bootstrapping is a method for constructing a (zero-coupon) fixed-income yield curve from the prices of a set of coupon-bearing products, e.g. bonds and swaps.
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Borel subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup.
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Central series
In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial.
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Characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots.
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Cholesky decomposition
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced /ʃ-/) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g. Monte Carlo simulations.
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Commuting matrices
In linear algebra, two matrices A and B are said to commute if AB.
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Complexification (Lie group)
In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a complex analytic homomorphism between the complex Lie groups.
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Computational complexity of mathematical operations
The following tables list the computational complexity of various algorithms for common mathematical operations.
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Condition number
In the field of numerical analysis, the condition number of a function with respect to an argument measures how much the output value of the function can change for a small change in the input argument.
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Cronbach's alpha
In statistics (classical test theory), Cronbach's \alpha (alpha) is the trivial name used for tau-equivalent reliability (\rho_T)Cho (2016), https://dx.doi.org/10.1177/1094428116656239 as a (lowerbound) estimate of the reliability of a psychometric test.
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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).
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Derivative of the exponential map
In the theory of Lie groups, the exponential map is a map from the Lie algebra of a Lie group into.
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Determinant
In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.
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Determinant identities
In mathematics the determinant is an operator which has certain useful identities.
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Diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.
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Diagonalizable matrix
In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P−1AP is a diagonal matrix.
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Divided differences
In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions.
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E8 lattice
In mathematics, the E8 lattice is a special lattice in R8.
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Eigendecomposition of a matrix
In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors.
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Eigenvalue algorithm
In numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix.
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Eigenvalues and eigenvectors
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.
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Elementary matrix
In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation.
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Engel's theorem
In representation theory, a branch of mathematics, Engel's theorem is one of the basic theorems in the theory of Lie algebras; it asserts that for a Lie algebra two concepts of nilpotency are identical.
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Factorization
In mathematics, factorization (also factorisation in some forms of British English) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
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Flag (linear algebra)
In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a finite-dimensional vector space V. Here "increasing" means each is a proper subspace of the next (see filtration): If we write the dim Vi.
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Gauss–Seidel method
In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a linear system of equations.
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Gaussian elimination
In linear algebra, Gaussian elimination (also known as row reduction) is an algorithm for solving systems of linear equations.
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Generalized flag variety
In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold.
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Generalized minimal residual method
In mathematics, the generalized minimal residual method (GMRES) is an iterative method for the numerical solution of a nonsymmetric system of linear equations.
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Givens rotation
In numerical linear algebra, a Givens rotation is a rotation in the plane spanned by two coordinates axes.
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Glossary of Lie algebras
This is a glossary for the terminology applied in the mathematical theories of Lie algebras.
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Gram–Schmidt process
In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalising a set of vectors in an inner product space, most commonly the Euclidean space Rn equipped with the standard inner product.
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Heisenberg group
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form \end under the operation of matrix multiplication.
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Hessenberg matrix
In linear algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular.
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Iliffe vector
In computer programming, an Iliffe vector, also known as a display, is a data structure used to implement multi-dimensional arrays.
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Incomplete Cholesky factorization
In numerical analysis, an incomplete Cholesky factorization of a symmetric positive definite matrix is a sparse approximation of the Cholesky factorization.
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Incomplete LU factorization
In numerical linear algebra, an incomplete LU factorization (abbreviated as ILU) of a matrix is a sparse approximation of the LU factorization often used as a preconditioner.
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Irreducibility (mathematics)
In mathematics, the concept of irreducibility is used in several ways.
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Iterative method
In computational mathematics, an iterative method is a mathematical procedure that uses an initial guess to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones.
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Iwasawa decomposition
In 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 product of an orthogonal matrix and an upper triangular matrix (a consequence of Gram–Schmidt orthogonalization).
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Joel Lee Brenner
Joel Lee Brenner (–) was an American mathematician who specialized in matrix theory, linear algebra, and group theory.
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Jordan matrix
In the mathematical discipline of matrix theory, a Jordan block over a ring R (whose identities are the zero 0 and one 1) is a matrix composed of 0 elements everywhere except for the diagonal, which is filled with a fixed element \lambda\in R, and for the superdiagonal, which is composed of ones.
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Jordan normal form
In linear algebra, a Jordan normal form (often called Jordan canonical form) of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis.
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Kalman filter
Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe.
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Keller's conjecture
In geometry, Keller's conjecture is the conjecture that in any tiling of Euclidean space by identical hypercubes there are two cubes that meet face to face.
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Kuiper's theorem
In mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H.
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LAPACK
LAPACK (Linear Algebra Package) is a standard software library for numerical linear algebra.
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Laplace expansion
In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression for the determinant |B| of an n × n matrix B that is a weighted sum of the determinants of n sub-matrices of B, each of size (n−1) × (n−1).
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Lattice (discrete subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure.
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Lexicostatistics
Lexicostatistics is a method of comparative linguistics that involves comparing the percentage of lexical cognates between languages to determine their relationship.
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Lie group decomposition
In mathematics, Lie group decompositions are used to analyse the structure of Lie groups and associated objects, by showing how they are built up out of subgroups.
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Lie–Kolchin theorem
In mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups; Lie's theorem is the analog for linear Lie algebras.
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Linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.
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Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices (under matrix multiplication) that is defined by polynomial equations.
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Linear least squares (mathematics)
In statistics and mathematics, linear least squares is an approach to fitting a mathematical or statistical model to data in cases where the idealized value provided by the model for any data point is expressed linearly in terms of the unknown parameters of the model.
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List of linear algebra topics
This is a list of linear algebra topics.
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List of matrices
This page lists some important classes of matrices used in mathematics, science and engineering.
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List of numerical analysis topics
This is a list of numerical analysis topics.
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List of Runge–Kutta methods
Runge–Kutta methods are methods for the numerical solution of the ordinary differential equation which take the form Each method listed on this page is defined by its Butcher tableau, which puts the coefficients of the method in a table as follows: \begin c_1 & a_ & a_& \dots & a_\\ c_2 & a_ & a_& \dots & a_\\ \vdots & \vdots & \vdots& \ddots& \vdots\\ c_s & a_ & a_& \dots & a_ \\ \hline \end.
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List of terms relating to algorithms and data structures
The NIST Dictionary of Algorithms and Data Structures is a reference work maintained by the U.S. National Institute of Standards and Technology.
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List of triangle topics
This list of triangle topics includes things related to the geometric shape, either abstractly, as in idealizations studied by geometers, or in triangular arrays such as Pascal's triangle or triangular matrices, or concretely in physical space.
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LU decomposition
In numerical analysis and linear algebra, LU decomposition (where "LU" stands for "lower–upper", and also called LU factorization) factors a matrix as the product of a lower triangular matrix and an upper triangular matrix.
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Matrix (mathematics)
In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
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Matrix analysis
In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties.
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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.
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Matrix group
In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication.
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Matrix ring
In abstract algebra, a matrix ring is any collection of matrices over some ring R that form a ring under matrix addition and matrix multiplication.
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Matrix splitting
In the mathematical discipline of numerical linear algebra, a matrix splitting is an expression which represents a given matrix as a sum or difference of matrices.
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Metabelian group
In mathematics, a metabelian group is a group whose commutator subgroup is abelian.
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Modular group
In mathematics, the modular group is the projective special linear group PSL(2,Z) of 2 x 2 matrices with integer coefficients and unit determinant.
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Molecular phylogenetics
Molecular phylogenetics is the branch of phylogeny that analyzes genetic, hereditary molecular differences, predominately in DNA sequences, to gain information on an organism's evolutionary relationships.
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Moore–Penrose inverse
In mathematics, and in particular linear algebra, a pseudoinverse of a matrix is a generalization of the inverse matrix.
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Nest algebra
In functional analysis, a branch of mathematics, nest algebras are a class of operator algebras that generalise the upper-triangular matrix algebras to a Hilbert space context.
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Newton polynomial
In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points.
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Nilpotent Lie algebra
In mathematics, a Lie algebra is nilpotent if its lower central series eventually becomes zero.
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Nilpotent matrix
In linear algebra, a nilpotent matrix is a square matrix N such that for some positive integer k. The smallest such k is sometimes called the index of N. More generally, a nilpotent transformation is a linear transformation L of a vector space such that Lk.
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Normal matrix
In mathematics, a complex square matrix is normal if where is the conjugate transpose of.
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Packed storage matrix
A packed storage matrix, also known as packed matrix, is a term used in programming for representing an m\times n matrix.
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Pascal matrix
In mathematics, particularly matrix theory and combinatorics, the Pascal matrix is an infinite matrix containing the binomial coefficients as its elements.
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Polyphase matrix
In signal processing, a polyphase matrix is a matrix whose elements are filter masks.
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Positive-definite matrix
In linear algebra, a symmetric real matrix M is said to be positive definite if the scalar z^Mz is strictly positive for every non-zero column vector z of n real numbers.
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Projective variety
In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective ''n''-space Pn over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety.
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QR algorithm
In numerical linear algebra, the QR algorithm is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix.
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QR decomposition
In linear algebra, a QR decomposition (also called a QR factorization) of a matrix is a decomposition of a matrix A into a product A.
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Regular chain
In computer algebra, a regular chain is a particular kind of triangular set in a multivariate polynomial ring over a field.
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Riordan array
A Riordan array is an infinite lower triangular matrix, D, constructed out of two formal power series, d(t) and h(t), in such a way that d_.
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Schur decomposition
In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition.
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Separation principle
In control theory, a separation principle, more formally known as a principle of separation of estimation and control, states that under some assumptions the problem of designing an optimal feedback controller for a stochastic system can be solved by designing an optimal observer for the state of the system, which feeds into an optimal deterministic controller for the system.
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Serial module
In abstract algebra, a uniserial module M is a module over a ring R, whose submodules are totally ordered by inclusion.
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Split Lie algebra
In the mathematical field of Lie theory, a split Lie algebra is a pair (\mathfrak, \mathfrak) where \mathfrak is a Lie algebra and \mathfrak is a splitting Cartan subalgebra, where "splitting" means that for all x \in \mathfrak, \operatorname_ x is triangularizable.
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Square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns.
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Stiff equation
In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small.
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Stirling number
In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems.
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Stirling numbers of the first kind
In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations.
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Stirling numbers of the second kind
In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of n objects into k non-empty subsets and is denoted by S(n,k) or \textstyle \lbrace\rbrace.
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Strict-feedback form
In control theory, dynamical systems are in strict-feedback form when they can be expressed as \dot_1.
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Successive over-relaxation
In numerical linear algebra, the method of successive over-relaxation (SOR) is a variant of the Gauss–Seidel method for solving a linear system of equations, resulting in faster convergence.
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Sylvester equation
In mathematics, in the field of control theory, a Sylvester equation is a matrix equation of the form: Then given matrices A,B, and C, the problem is to find the possible matrices X that obey this equation.
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Sylvester's criterion
In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite.
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Trace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e., where aii denotes the entry on the ith row and ith column of A. The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis.
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Triangular array
In mathematics and computing, a triangular array of numbers, polynomials, or the like, is a doubly indexed sequence in which each row is only as long as the row's own index.
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Triangular matrix
In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix.
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Triangular network coding
In coding theory, triangular network coding (TNC) is a network coding based packet coding scheme introduced by.
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Triangulation (disambiguation)
Triangulation refers to measurement by using triangles, or angle measurements in surveying.
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Vectorization (mathematics)
In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a column vector.
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Volodin space
In mathematics, more specifically in topology, the Volodin space X of a ring R is a subspace of the classifying space BGL(R) given by where U_n(R) \subset GL_n(R) is the subgroup of upper triangular matrices with 1's on the diagonal (i.e., the unipotent radical of the standard Borel) and \sigma a permutation matrix thought of as an element in GL_n(R) and acting (superscript) by conjugation.
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Weight (representation theory)
In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one-dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group.
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Zappa–Szép product
In mathematics, especially group theory, the Zappa–Szép product (also known as the Zappa–Rédei-Szép product, general product, knit product or exact factorization) describes a way in which a group can be constructed from two subgroups.
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References
[1] https://en.wikipedia.org/wiki/Triangular_matrix
