• list of matrices
• glossary of tensor theory
  Glossary of tensor theory
  This is a glossary of tensor theory. For expositions of tensor theory from different points of view, see:* Tensor* Tensor * Application of tensor theory in engineering science...
  
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Linear equation

Linear equation

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....

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• System of linear equations
• Determinant
  Determinant
  In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
  
  
  • Minor
    Minor (linear algebra)
    In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns...
    
    
  • Cauchy–Binet formula
• Cramer's rule
  Cramer's rule
  In linear algebra, Cramer's rule is a theorem, which gives an expression for the solution of a system of linear equations with as many equations as unknowns, valid in those cases where there is a unique solution...
  
  
• Gaussian elimination
  Gaussian elimination
  In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...
  
  
• Gauss–Jordan elimination
  Gauss–Jordan elimination
  In linear algebra, Gauss–Jordan elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations. It is a variation of Gaussian elimination. Gaussian elimination places zeros below each pivot in the matrix, starting with the top row and working downwards....
  
  
• Strassen algorithm
  Strassen algorithm
  In the mathematical discipline of linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm used for matrix multiplication...
  
  

Matrices

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

• Matrix theory
• Matrix addition
  Matrix addition
  In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations which could also be considered as a kind of addition for matrices, the direct sum and the Kronecker sum....
  
  
• Matrix multiplication
  Matrix multiplication
  In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
  
  
• Basis transformation matrix
• Characteristic polynomial
  Characteristic polynomial
  In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....
  
  
• Trace
• Eigenvalue, eigenvector and eigenspace
  Eigenvalue, eigenvector and eigenspace
  The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix...
  
  
  • Cayley–Hamilton theorem
    Cayley–Hamilton theorem
    In linear algebra, the Cayley–Hamilton theorem states that every square matrix over a commutative ring satisfies its own characteristic equation....
    
    
  • Spread of a matrix
    Spread of a matrix
    In matrix theory, the spread of a matrix describes how far apart the eigenvalues are in the complex plane.Suppose A is a square matrix with eigenvalues \lambda_1, \ldots, \lambda_n...
    
    
• Jordan normal form
  Jordan normal form
  In linear algebra, a Jordan normal form of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called Jordan matrix, representing the operator on some basis...
  
  
• Rank
• Matrix inversion, invertible matrix
  • Pseudoinverse
    Pseudoinverse
    In mathematics, and in particular linear algebra, a pseudoinverse of a matrix is a generalization of the inverse matrix. The most widely known type of matrix pseudoinverse is the Moore–Penrose pseudoinverse, which was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951 and...
    
    
• Adjugate
• Transpose
  Transpose
  In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
  
  
  • Dot product
    Dot product
    In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
    
    
  • Symmetric matrix
  • Matrix congruence
    Matrix congruence
    In mathematics, two matrices A and B over a field are called congruent if there exists an invertible matrix P over the same field such thatwhere "T" denotes the matrix transpose...
    
    
    • Congruence relation
  • Orthogonal matrix
    Orthogonal matrix
    In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
    
    
  • Skew-symmetric matrix
    Skew-symmetric matrix
    In mathematics, and in particular linear algebra, a skew-symmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation If the entry in the and is aij, i.e...
    
    
  • Conjugate transpose
    Conjugate transpose
    In mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry...
    
    
    • Unitary matrix
    • Hermitian matrix, Antihermitian
• Positive definite, positive semidefinite
  Positive semidefinite
  In mathematics, positive semidefinite may refer to:* positive-semidefinite matrix* positive-semidefinite function...
  
  , positive-definite matrix
  Positive-definite matrix
  In linear algebra, a positive-definite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positive-definite symmetric bilinear form ....
  
  
• Pfaffian
  Pfaffian
  In mathematics, the determinant of a skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries. This polynomial is called the Pfaffian of the matrix, The term Pfaffian was introduced by who named them after Johann Friedrich Pfaff...
  
  
• Projection
  Projection (linear algebra)
  In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged....
  
  
• 2 × 2 real matrices
• Similar matrix
• Spectral theorem
  Spectral theorem
  In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...
  
  
• Woodbury matrix identity
  Woodbury matrix identity
  In mathematics , the Woodbury matrix identity, named after Max A. Woodbury says that the inverse of a rank-k correction of some matrix can be computed by doing a rank-k correction to the inverse of the original matrix...
  
  
• Perron–Frobenius theorem
  Perron–Frobenius theorem
  In linear algebra, the Perron–Frobenius theorem, proved by and , asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector has strictly positive components, and also asserts a similar statement for certain classes of...
  
  
• List of matrices
  • Diagonal matrix
    Diagonal matrix
    In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...
    
    , main diagonal
    • Diagonalizable matrix
      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...
      
      
  • Triangular matrix
    Triangular matrix
    In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where either all the entries below or all the entries above the main diagonal are zero...
    
    
  • Tridiagonal matrix
  • Block matrix
    Block matrix
    In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a matrix broken into sections called blocks. Looking at it another way, the matrix is written in terms of smaller matrices. We group the rows and columns into adjacent 'bunches'. A partition is the rectangle...
    
    
  • Sparse matrix
    Sparse matrix
    In the subfield of numerical analysis, a sparse matrix is a matrix populated primarily with zeros . The term itself was coined by Harry M. Markowitz....
    
    
  • Hessenberg matrix
    Hessenberg matrix
    In 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 entries below the first subdiagonal, and a lower Hessenberg matrix has zero entries above the first superdiagonal...
    
    
  • Hessian matrix
    Hessian matrix
    In mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...
    
    
  • Vandermonde matrix
  • Stochastic matrix
    Stochastic matrix
    In mathematics, a stochastic matrix is a matrix used to describe the transitions of a Markov chain. It has found use in probability theory, statistics and linear algebra, as well as computer science...
    
    
  • Toeplitz matrix
    Toeplitz matrix
    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 right is constant...
    
    
    • Circulant matrix
      Circulant matrix
      In linear algebra, a circulant matrix is a special kind of Toeplitz matrix where each row vector is rotated one element to the right relative to the preceding row vector. In numerical analysis, circulant matrices are important because they are diagonalized by a discrete Fourier transform, and hence...
      
      
  • Hankel matrix
  • (0,1)-matrix

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 :...

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• Cholesky decomposition
  Cholesky decomposition
  In linear algebra, the Cholesky decomposition or Cholesky triangle is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. It was discovered by André-Louis Cholesky for real matrices...
  
  
• LU decomposition
  LU decomposition
  In linear algebra, LU decomposition is a matrix decomposition which writes a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. This decomposition is used in numerical analysis to solve systems of linear...
  
  
• QR decomposition
  QR decomposition
  In linear algebra, a QR decomposition of a matrix is a decomposition of a matrix A into a product A=QR of an orthogonal matrix Q and an upper triangular matrix R...
  
  
• Spectral theorem
  Spectral theorem
  In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...
  
  
• Singular value decomposition
  Singular value decomposition
  In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....
  
  
  • Higher-order singular value decomposition
• Schur decomposition
  Schur decomposition
  In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition.- Statement :...
  
  
  • Schur complement
    Schur complement
    In linear algebra and the theory of matrices,the Schur complement of a matrix block is defined as follows.Suppose A, B, C, D are respectivelyp×p, p×q, q×p...
    
    
  • Haynsworth inertia additivity formula
    Haynsworth inertia additivity formula
    In mathematics, the Haynsworth inertia additivity formula, discovered by Emilie Virginia Haynsworth , concerns the number of positive, negative, and zero eigenvalues of a Hermitian matrix and of block matrices into which it is partitioned....
    
    

Computations

• Householder transformation
  Householder transformation
  In linear algebra, a Householder transformation is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. Householder transformations are widely used in numerical linear algebra, to perform QR decompositions and in the first step of the QR algorithm...
  
  
• Least squares
  Least squares
  The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
  
  , linear least squares
  Linear least squares
  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...
  
  
• Gram–Schmidt process
  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...
  
  

Vectors

• Scalar multiplication
  Scalar multiplication
  In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra . In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction...
  
  
• Linear combination
  Linear combination
  In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
  
  
• Linear span
  Linear span
  In the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...
  
  
• Linear independence
  Linear independence
  In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent...
  
  
• Basis
  Basis (linear algebra)
  In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
  
  
• Coordinates vector

Vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

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• Basis
  Basis (linear algebra)
  In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
  
  
  • Change of basis
    Change of basis
    In linear algebra, change of basis refers to the conversion of vectors and linear transformations between matrix representations which have different bases.-Expression of a basis:...
    
    
  • Hamel basis
• Dimension theorem for vector spaces
  Dimension theorem for vector spaces
  In mathematics, the dimension theorem for vector spaces states that all bases of a vector space have equally many elements. This number of elements may be finite, or given by an infinite cardinal number, and defines the dimension of the space....
  
  
  • Hamel dimension
• Examples of vector spaces
  Examples of vector spaces
  This page lists some examples of vector spaces. See vector space for the definitions of terms used on this page. See also: dimension, basis.Notation. We will let F denote an arbitrary field such as the real numbers R or the complex numbers C...
  
  
• Linear map
  • Shear mapping or Galilean transformation
    Galilean transformation
    The Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. This is the passive transformation point of view...
    
    
  • Squeeze mapping
    Squeeze mapping
    In linear algebra, a squeeze mapping is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a Euclidean motion.For a fixed positive real number r, the mapping →...
    
     or Lorentz transformation
    Lorentz transformation
    In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
    
    
• Column space
  Column space
  In linear algebra, the column space of a matrix is the set of all possible linear combinations of its column vectors. The column space of an m × n matrix is a subspace of m-dimensional Euclidean space...
  
  
• Row space
  Row space
  In linear algebra, the row space of a matrix is the set of all possible linear combinations of its row vectors. The row space of an m × n matrix is a subspace of n-dimensional Euclidean space...
  
  
• Null space
  Null space
  In linear algebra, the kernel or null space of a matrix A is the set of all vectors x for which Ax = 0. The kernel of a matrix with n columns is a linear subspace of n-dimensional Euclidean space...
  
  , nullity
  Nullity
  Nullity may refer to:* Nullity , a legal declaration that no marriage had ever come into being* Nullity, the dimension of the null space of a mathematical operator or matrix...
  
  
• Rank-nullity theorem
  Rank-nullity theorem
  In mathematics, the rank–nullity theorem of linear algebra, in its simplest form, states that the rank and the nullity of a matrix add up to the number of columns of the matrix. Specifically, if A is an m-by-n matrix over some field, thenThis applies to linear maps as well...
  
  
• Nullity theorem
  Nullity theorem
  The nullity theorem is a mathematical theorem about the inverse of a partitioned matrix, which states that the nullity of a block in a matrix equals the nullity of the complementary block in its inverse matrix. Here, the nullity is the dimension of the kernel...
  
  
• Dual space
  Dual space
  In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
  
  
  • Linear function
    Linear function
    In mathematics, the term linear function can refer to either of two different but related concepts:* a first-degree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....
    
    
  • Linear functional
    Linear functional
    In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars.  In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
    
    
• Orthogonality
  Orthogonality
  Orthogonality occurs when two things can vary independently, they are uncorrelated, or they are perpendicular.-Mathematics:In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle...
  
  
• Orthogonal complement
• Orthogonal projection
• Outer product
  Outer product
  In linear algebra, the outer product typically refers to the tensor product of two vectors. The result of applying the outer product to a pair of vectors is a matrix...
  
  
• Improper rotation
  Improper rotation
  In 3D geometry, an improper rotation, also called rotoreflection or rotary reflection is, depending on context, a linear transformation or affine transformation which is the combination of a rotation about an axis and a reflection in a plane perpendicular to the axis.Equivalently it is the...
  
  
• Category of vector spaces
  Category of vector spaces
  In mathematics, especially category theory, the category K-Vect has all vector spaces over a fixed field K as objects and K-linear transformations as morphisms...
  
  
• Linear subspace
  Linear subspace
  The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
  
  
  • Euclidean subspace
    Euclidean subspace
    In linear algebra, a Euclidean subspace is a set of vectors that is closed under addition and scalar multiplication. Geometrically, a subspace is a flat in n-dimensional Euclidean space that passes through the origin...
    
    
• Normed vector space
  Normed vector space
  In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....
  
  
• Inner product space
  Inner product space
  In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...
  
  

Multilinear algebra

Multilinear algebra

In mathematics, multilinear algebra extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of p-vectors and multivectors with Grassmann algebra.-Origin:In a vector space...

• Tensor
  Tensor
  Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
  
  
  • Classical treatment of tensors
  • Intermediate treatment of tensors
  • Component-free treatment of tensors
• Tensor algebra
  Tensor algebra
  In mathematics, the tensor algebra of a vector space V, denoted T or T•, is the algebra of tensors on V with multiplication being the tensor product...
  
  
  • Exterior algebra
    Exterior algebra
    In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...
    
    
  • Symmetric algebra
    Symmetric algebra
    In mathematics, the symmetric algebra S on a vector space V over a field K is the free commutative unital associative algebra over K containing V....
    
    
  • Clifford algebra
    Clifford algebra
    In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
    
    
  • Geometric algebra
    Geometric algebra
    Geometric algebra , together with the associated Geometric calculus, provides a comprehensive alternative approach to the algebraic representation of classical, computational and relativistic geometry. GA now finds application in all of physics, in graphics and in robotics...