This is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector spaces and through matrices.

Linear equations

Linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...

System of linear equations In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variable (math), variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three ...

* Determinant **

Minor Minor may refer to: * Minor (law), a person under the age of certain legal activities. ** A person who has not reached the age of majority * Academic minor, a secondary field of study in undergraduate education Music theory *Minor chord ** Barb ...

Cauchy–Binet formula In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so tha ...

Cramer's rule In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants o ...

Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...

* Gauss–Jordan elimination *

Overcompleteness Overcompleteness is a concept from linear algebra that is widely used in mathematics, computer science, engineering, and statistics (usually in the form of overcomplete frames). It was introduced by R. J. Duffin and A. C. Schaeffer in 1952. For ...

Strassen algorithm In linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm for matrix multiplication. It is faster than the standard matrix multiplication algorithm for large matrices, with a better asymptotic complexity, although t ...

Matrices

Matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

* Matrix addition * Matrix multiplication * Basis transformation matrix *

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. It has the determinant and the trace of the matrix among its coefficients. The chara ...

Trace Trace may refer to: Arts and entertainment Music * Trace (Son Volt album), ''Trace'' (Son Volt album), 1995 * Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * The Trace (album), ''The ...

* Eigenvalue, eigenvector and eigenspace ** Cayley–Hamilton theorem ** Spread of a matrix ** Jordan normal form **

Weyr canonical form In mathematics, in linear algebra, a Weyr canonical form (or, Weyr form or Weyr matrix) is a square matrix satisfying certain conditions. A square matrix is said to be ''in'' the Weyr canonical form if the matrix satisfies the conditions defining ...

* Rank * Matrix inversion, invertible matrix ** Pseudoinverse *

Adjugate In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix and is denoted by . It is also occasionally known as adjunct matrix, or "adjoint", though the latter today normally refers to a differe ...

* Transpose ** Dot product ** Symmetric matrix ** Orthogonal matrix **

Skew-symmetric matrix In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if a_ ...

** Conjugate transpose *** Unitary matrix *** Hermitian matrix, Antihermitian matrix *

Positive-definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite fu ...

, positive-semidefinite matrix * Pfaffian *

Projection Projection, projections or projective may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphic ...

* Spectral theorem *

Perron–Frobenius theorem In matrix theory, 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 can be chosen to have strictly positive component ...

List of matrices This article lists some important classes of matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called ''entries''. Matrices have a long history of both st ...

** Diagonal matrix, main diagonal *** Diagonalizable matrix ** Triangular matrix ** Tridiagonal matrix **

Block matrix In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original mat ...

** Sparse matrix ** Hessenberg matrix ** Hessian matrix ** Vandermonde matrix ** Stochastic matrix ** Toeplitz matrix *** Circulant matrix ** Hankel matrix **

(0,1)-matrix A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1) matrix is a matrix with entries from the Boolean domain Such a matrix can be used to represent a binary relation between a pair of finite sets. Matrix representati ...

Matrix decompositions

Matrix decomposition * Cholesky decomposition * LU decomposition * QR decomposition * Polar decomposition *

Reducing subspace In linear algebra, a reducing subspace W of a linear map T:V\to V from a Hilbert space V to itself is an invariant subspace of T whose orthogonal complement W^\perp is also an invariant subspace of T. That is, T(W) \subseteq W and T(W^\perp) \subse ...

* Spectral theorem * Singular value decomposition **

Higher-order singular value decomposition In multilinear algebra, the higher-order singular value decomposition (HOSVD) of a tensor is a specific orthogonal Tucker decomposition. It may be regarded as one generalization of the matrix singular value decomposition. It has applications in co ...

Schur decomposition In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily equivalent to an upper tria ...

Schur complement In linear algebra and the theory of matrices, the Schur complement of a block matrix is defined as follows. Suppose ''p'', ''q'' are nonnegative integers, and suppose ''A'', ''B'', ''C'', ''D'' are respectively ''p'' × ''p'', ''p'' × ''q'', ''q'' ...

Haynsworth inertia additivity formula In mathematics, the Haynsworth inertia additivity formula, discovered by Emilie Virginia Haynsworth (1916–1985), concerns the number of positive, negative, and zero eigenvalues of a Hermitian matrix and of block matrices into which it is partiti ...

Relations

* Matrix equivalence * Matrix congruence * Matrix similarity *

Matrix consimilarity In linear algebra, two ''n''-by-''n'' matrices ''A'' and ''B'' are called consimilar if : A = S B \bar^ \, for some invertible n \times n matrix S, where \bar denotes the elementwise complex conjugation. So for real matrices similar by some real ...

Row equivalence In linear algebra, two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. Alternatively, two ''m'' × ''n'' matrices are row equivalent if and only if they have the same row ...

Computations

Elementary row operations In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GL''n''(F) when F is a field. Left multiplication (pre-multi ...

* Householder transformation *

Least squares The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...

linear least squares Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and ...

* Gram–Schmidt process * Woodbury matrix identity

Vector spaces

Vector space * Linear combination *

Linear span In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized ...

* Linear independence * Scalar multiplication * Basis ** Change of basis ** Hamel basis * Cyclic decomposition theorem * Dimension theorem for vector spaces ** Hamel dimension *

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''. Let ''F'' denote an arbitrary field such as the real numbers R or the complex numbers C. T ...

* Linear map ** Shear mapping or

Galilean transformation In physics, a 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. These transformations together with spatial rotatio ...

** Squeeze mapping or Lorentz transformation *

Linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...

** Row and column spaces ** Column space ** Row space **

Cyclic subspace In mathematics, in linear algebra and functional analysis, a cyclic subspace is a certain special subspace of a vector space associated with a vector in the vector space and a linear transformation of the vector space. The cyclic subspace associate ...

** Null space, nullity ** Rank–nullity theorem ** Nullity theorem *

Dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...

** Linear function ** Linear functional * Category of vector spaces

Structures

* Topological vector space * Normed vector space * Inner product space ** Euclidean space **

Orthogonality In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

** Orthogonal complement ** Orthogonal projection **

Orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...

* Pseudo-Euclidean space ** Null vector ** Indefinite orthogonal group *

Orientation (geometry) In geometry, the orientation, angular position, attitude, bearing, or direction of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it occupies. More specifically, it refers to the imagin ...

** Improper rotation * Symplectic structure

Multilinear algebra

Multilinear algebra * Tensor **

Classical treatment of tensors In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...

** Component-free treatment of tensors **

Gamas's Theorem Gamas's theorem is a result in multilinear algebra which states the necessary and sufficient conditions for a tensor symmetrized by an irreducible representation of the symmetric group S_n to be zero. It was proven in 1988 by Carlos Gamas. Additi ...

Outer product In linear algebra, the outer product of two coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An ea ...

* Tensor algebra ** Exterior algebra ** Symmetric algebra **

Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...

Geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ge ...

Topics related to affine spaces

Affine space *

Affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...

* Affine group * Affine geometry *

Affine coordinate system In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...

* Flat (geometry) *

Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

Euclidean group In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). ...

* Poincaré group *

Galilean group In physics, a 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. These transformations together with spatial rotatio ...

Projective space

Projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

* Projective transformation * Projective geometry * Projective linear group * Quadric and conic section