Frobenius normal form
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 obtained by conjugation by invertible matrices o
Dual norm
In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space.
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. Specifically, the vectorization
Stabilizer code
The theory of quantum error correction plays a prominent role in the practical realization and engineering ofquantum computing and quantum communication devices. The first quantumerror-correcting code
Finite von Neumann algebra
In mathematics, a finite von Neumann algebra is a von Neumann algebra in which every isometry is a unitary. In other words, for an operator V in a finite von Neumann algebra if , then . In terms of th
Multilinear form
In abstract algebra and multilinear algebra, a multilinear form on a vector space over a field is a map that is separately K-linear in each of its k arguments. More generally, one can define multiline
Leibniz formula for determinants
In algebra, the Leibniz formula, named in honor of Gottfried Leibniz, expresses the determinant of a square matrix in terms of permutations of the matrix elements. If is an matrix, where is the entry
Loewner order
In mathematics, Loewner order is the partial order defined by the convex cone of positive semi-definite matrices. This order is usually employed to generalize the definitions of monotone and concave/c
Integer points in convex polyhedra
The study of integer points in convex polyhedra is motivated by questions such as "how many nonnegative integer-valued solutions does a system of linear equations with nonnegative coefficients have" o
Semilinear map
In linear algebra, particularly projective geometry, a semilinear map between vector spaces V and W over a field K is a function that is a linear map "up to a twist", hence semi-linear, where "twist"
Defective matrix
In linear algebra, a defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable. In particular, an n × n matrix is defective if and on
Orientation (vector space)
The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the thre
Canonical basis
In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context:
* In a coordinate space, and more generally in a free module,
Dual basis
In linear algebra, given a vector space V with a basis B of vectors indexed by an index set I (the cardinality of I is the dimension of V), the dual set of B is a set B∗ of vectors in the dual space V
Generalized eigenvector
In linear algebra, a generalized eigenvector of an matrix is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. Let be an -dimensional vector sp
Cartesian tensor
In geometry and linear algebra, a Cartesian tensor uses an orthonormal basis to represent a tensor in a Euclidean space in the form of components. Converting a tensor's components from one such basis
Combinatorial matrix theory
Combinatorial matrix theory is a branch of linear algebra and combinatorics that studies matrices in terms of the patterns of nonzeros and of positive and negative values in their coefficients. Concep
Joint spectral radius
In mathematics, the joint spectral radius is a generalization of the classical notion of spectral radius of a matrix, to sets of matrices. In recent years this notion has found applications in a large
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of tw
Majorization
In mathematics, majorization is a preorder on vectors of real numbers. For a vector , we denote by the vector with the same components, but sorted in descending order. Given , we say that weakly major
Sherman–Morrison formula
In mathematics, in particular linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of the sum of an invertible matrix and the outer pro
Pseudoscalar
In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not. Any scalar product between a pseudovector
Matrix Chernoff bound
For certain applications in linear algebra, it is useful to know properties of the probability distribution of the largest eigenvalue of a finite sum of random matrices. Suppose is a finite sequence o
Reduction (mathematics)
In mathematics, reduction refers to the rewriting of an expression into a simpler form. For example, the process of rewriting a fraction into one with the smallest whole-number denominator possible (w
Restricted isometry property
In linear algebra, the restricted isometry property (RIP) characterizes matrices which are nearly orthonormal, at least when operating on sparse vectors. The concept was introduced by Emmanuel Candès
Orthogonality (mathematics)
In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity to the linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B are
Computing the permanent
In linear algebra, the computation of the permanent of a matrix is a problem that is thought to be more difficult than the computation of the determinant of a matrix despite the apparent similarity of
Scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar
Coates graph
In mathematics, the Coates graph or Coates flow graph, named after C.L. Coates, is a graph associated with the Coates' method for the solution of a system of linear equations. The Coates graph Gc(A) a
Line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is
Quaternionic vector space
In mathematics, a left (or right) quaternionic vector space is a left (or right) H-module where H is the (non-commutative) division ring of quaternions. The space Hn of n-tuples of quaternions is both
Change of basis
In mathematics, an ordered basis of a vector space of finite dimension n allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of n scalars called co
Dual space
In mathematics, any vector space has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on , together with the vector space structure of pointwise addition
Spherical basis
In pure and applied mathematics, particularly quantum mechanics and computer graphics and their applications, a spherical basis is the basis used to express spherical tensors. The spherical basis clos
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for whose vectors are orthonormal, that is, they are all unit vectors an
Linear relation
In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution. More precisely, if are elements of
Hurwitz determinant
In mathematics, Hurwitz determinants were introduced by Adolf Hurwitz, who used them to give a criterion for all roots of a polynomial to have negative real part.
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in
Pairing
In mathematics, a pairing is an R-bilinear map from the Cartesian product of two R-modules, where the underlying ring R is commutative.
Pohlke's theorem
Pohlke's theorem is the fundamental theorem of axonometry. It was established 1853 by the German painter and teacher of descriptive geometry Karl Wilhelm Pohlke. The first proof of the theorem was pub
Rayleigh theorem for eigenvalues
In mathematics, the Rayleigh theorem for eigenvalues pertains to the behavior of the solutions of an eigenvalue equation as the number of basis functions employed in its resolution increases. Rayleigh
K-SVD
In applied mathematics, K-SVD is a dictionary learning algorithm for creating a dictionary for sparse representations, via a singular value decomposition approach. K-SVD is a generalization of the k-m
Orthonormality
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in
Constant-recursive sequence
In mathematics and theoretical computer science, a constant-recursive sequence is an infinite sequence of numbers where each number in the sequence is equal to a fixed linear combination of one or mor
Ridge regression
Ridge regression is a method of estimating the coefficients of multiple-regression models in scenarios where the independent variables are highly correlated. It has been used in many fields including
Levi-Civita symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a
Basis (linear algebra)
In mathematics, a set B of vectors in a vector space V is called a basis if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of this
Mathcad
Mathcad is computer software for the verification, validation, documentation and re-use of mathematical calculations in engineering and science, notably mechanical, chemical, electrical, and civil eng
Resolvent set
In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important r
Gershgorin circle theorem
In mathematics, the Gershgorin circle theorem may be used to bound the spectrum of a square matrix. It was first published by the Soviet mathematician Semyon Aronovich Gershgorin in 1931. Gershgorin's
Special linear group
In mathematics, the special linear group SL(n, F) of degree n over a field F is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inv
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinit
S-procedure
The S-procedure or S-lemma is a mathematical result that gives conditions under which a particular quadratic inequality is a consequence of another quadratic inequality. The S-procedure was developed
Affine space
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
Total set
In functional analysis, a total set (also called a complete set) in a vector space is a set of linear functionals with the property that if a vector satisfies for all then is the zero vector. In a mor
Skew-Hamiltonian matrix
In linear algebra, skew-Hamiltonian matrices are special matrices which correspond to skew-symmetric bilinear forms on a symplectic vector space. Let V be a vector space, equipped with a symplectic fo
Kernel (algebra)
In algebra, the kernel of a homomorphism (function that preserves the structure) is generally the inverse image of 0 (except for groups whose operation is denoted multiplicatively, where the kernel is
Complex conjugate of a vector space
In mathematics, the complex conjugate of a complex vector space is a complex vector space , which has the same elements and additive group structure as but whose scalar multiplication involves conjuga
Independent equation
An independent equation is an equation in a system of simultaneous equations which cannot be derived algebraically from the other equations. The concept typically arises in the context of linear equat
Semisimple operator
In mathematics, a linear operator T on a vector space is semisimple if every T-invariant subspace has a complementary T-invariant subspace; in other words, the vector space is a semisimple representat
Rank (linear algebra)
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 number of linearly independent columns of A. This,
Generalizations of Pauli matrices
In mathematics and physics, in particular quantum information, the term generalized Pauli matrices refers to families of matrices which generalize the (linear algebraic) properties of the Pauli matric
Jordan–Chevalley decomposition
In mathematics, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley, expresses a linear operator as the sum of its commuting semisimple part and its nilpotent part. The
Nullspace property
In compressed sensing, the nullspace property gives necessary and sufficient conditions on the reconstruction of sparse signals using the techniques of -relaxation. The term "nullspace property" origi
Pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector i
Tensor product of Hilbert spaces
In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilb
Orthogonal diagonalization
In linear algebra, an orthogonal diagonalization of a symmetric matrix is a diagonalization by means of an orthogonal change of coordinates. The following is an orthogonal diagonalization algorithm th
Category of modules
In algebra, given a ring R, the category of left modules over R is the category whose objects are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For e
Gram–Schmidt process
In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean sp
Linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually
Split-complex number
In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components x and y, and is written z = x + y j, where j2 = 1. The conjugate of z is z∗
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plott
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
Rayleigh quotient
In mathematics, the Rayleigh quotient (/ˈreɪ.li/) for a given complex Hermitian matrix M and nonzero vector x is defined as: For real matrices and vectors, the condition of being Hermitian reduces to
Row and column spaces
In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the imag
Star domain
In geometry, a set in the Euclidean space is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an such that for all the line segment from to lies in Thi
Unitary element
In mathematics, an element x of a *-algebra is unitary if it satisfies In functional analysis, a linear operator A from a Hilbert space into itself is called unitary if it is invertible and its invers
Trace (linear algebra)
In linear algebra, the trace of a square matrix A, denoted tr(A), is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of A. The trace is only defined for
Hilbert–Poincaré series
In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series (also known under the name Hilbert series), named after David Hilbert and Henri Poincaré, is an adaptation of the n
Orthogonal basis
In mathematics, particularly linear algebra, an orthogonal basis for an inner product space is a basis for whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized,
Orthogonal transformation
In linear algebra, an orthogonal transformation is a linear transformation T : V → V on a real inner product space V, that preserves the inner product. That is, for each pair u, v of elements of V, we
Orthographic projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in wh
Linear complementarity problem
In mathematical optimization theory, the linear complementarity problem (LCP) arises frequently in computational mechanics and encompasses the well-known quadratic programming as a special case. It wa
Peetre's inequality
In mathematics, Peetre's inequality, named after Jaak Peetre, says that for any real number t and any vectors x and y in Rn, the following inequality holds:
Reducing subspace
In linear algebra, a reducing subspace of a linear map from a Hilbert space to itself is an invariant subspace of whose orthogonal complement is also an invariant subspace of That is, and One says tha
Bendixson's inequality
In mathematics, Bendixson's inequality is a quantitative result in the field of matrices derived by Ivar Bendixson in 1902. The inequality puts limits on the imaginary parts of Characteristic roots (e
Regularized least squares
Regularized least squares (RLS) is a family of methods for solving the least-squares problem while using regularization to further constrain the resulting solution. RLS is used for two main reasons. T
Symplectic vector space
In mathematics, a symplectic vector space is a vector space V over a field F (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping ω : V ×
Big M method
In operations research, the Big M method is a method of solving linear programming problems using the simplex algorithm. The Big M method extends the simplex algorithm to problems that contain "greate
Orthant
In geometry, an orthant or hyperoctant is the analogue in n-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions. In general an orthant in n-dimensions can be consid
Schmidt decomposition
In linear algebra, the Schmidt decomposition (named after its originator Erhard Schmidt) refers to a particular way of expressing a vector in the tensor product of two inner product spaces. It has num
Conformable matrix
In mathematics, a matrix is conformable if its dimensions are suitable for defining some operation (e.g. addition, multiplication, etc.).
Frame (linear algebra)
In linear algebra, a frame of an inner product space is a generalization of a basis of a vector space to sets that may be linearly dependent. In the terminology of signal processing, a frame provides
Homogeneous function
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called
Shear mapping
In plane geometry, a shear mapping is a linear map that displaces each point in a fixed direction, by an amount proportional to its signed distance from the line that is parallel to that direction and
Backus–Gilbert method
In mathematics, the Backus–Gilbert method, also known as the optimally localized average (OLA) method is named for its discoverers, geophysicists George E. Backus and James Freeman Gilbert. It is a re
Elementary matrix
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 GLn(F) when F
Linear equation over a ring
In algebra, linear equations and systems of linear equations over a field are widely studied. "Over a field" means that the coefficients of the equations and the solutions that one is looking for belo
Shear matrix
In mathematics, a shear matrix or transvection is an elementary matrix that represents the addition of a multiple of one row or column to another. Such a matrix may be derived by taking the identity m
Adjugate matrix
In linear algebra, the adjugate or classical adjoint of a square matrix A is the transpose of its cofactor matrix and is denoted by adj(A). It is also occasionally known as adjunct matrix, or "adjoint
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
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 variables. For example, is a system of three equations in the three v
Null vector
In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero element x of X for which q(x) = 0. In the theory of real bil
Normal element
In mathematics, an element x of a *-algebra is normal if it satisfies This definition stems from the definition of a normal linear operator in functional analysis, where a linear operator A from a Hil
Dieudonné determinant
In linear algebra, the Dieudonné determinant is a generalization of the determinant of a matrix to matrices over division rings and local rings. It was introduced by Dieudonné. If K is a division ring
Hermite normal form
In linear algebra, the Hermite normal form is an analogue of reduced echelon form for matrices over the integers Z. Just as reduced echelon form can be used to solve problems about the solution to the
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
Nonnegative rank (linear 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 that certain coefficients and entries of vect
Vector-valued function
A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. T
Asymmetric norm
In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm.
Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation from a vector space to itself (an endomorphism) such that . That is, whenever is applied twice to any vector, it give
Rule of Sarrus
In linear algebra, the Rule of Sarrus is a mnemonic device for computing the determinant of a matrix named after the French mathematician Pierre Frédéric Sarrus. Consider a matrix then its determinant
Lattice reduction
In mathematics, the goal of lattice basis reduction is to find a basis with short, nearly orthogonal vectors when given an integer lattice basis as input. This is realized using different algorithms,
Eigenvalues and eigenvectors
In linear algebra, an eigenvector (/ˈaɪɡənˌvɛktər/) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is a
Quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, is a quadratic form in the variables x and y. The coef
Tensor operator
In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical te
Line–line intersection
In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or another line. Distinguishing these cases and finding the intersection have uses, for example, in computer
Locally finite operator
In mathematics, a linear operator is called locally finite if the space is the union of a family of finite-dimensional -invariant subspaces. In other words, there exists a family of linear subspaces o
Dependence relation
In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence. Let be a set. A (binary) relation between an element of and a subset of is called a depe
Matrix difference equation
A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related to its own value at one or more previous poin
Overdetermined system
In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns. An overdetermined system is almost always inconsistent (it has no solution) when construct
Orthogonal complement
In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W⊥ of all vectors in V th
Triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits
Underdetermined system
In mathematics, a system of linear equations or a system of polynomial equations is considered underdetermined if there are fewer equations than unknowns (in contrast to an overdetermined system, wher
Antiunitary operator
In mathematics, an antiunitary transformation, is a bijective antilinear map between two complex Hilbert spaces such that for all and in , where the horizontal bar represents the complex conjugate. If
Graded (mathematics)
In mathematics, the term “graded” has a number of meanings, mostly related: In abstract algebra, it refers to a family of concepts:
* An algebraic structure is said to be -graded for an index set if
Projection-valued measure
In mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed
Lapped transform
In signal processing, a lapped transform is a type of linear discrete block transformation where the basis functions of the transformation overlap the block boundaries, yet the number of coefficients
Amitsur–Levitzki theorem
In algebra, the Amitsur–Levitzki theorem states that the algebra of n × n matrices over a commutative ring satisfies a certain identity of degree 2n. It was proved by Amitsur and Levitsky. In particul
Spinors in three dimensions
In mathematics, the spinor concept as specialised to three dimensions can be treated by means of the traditional notions of dot product and cross product. This is part of the detailed algebraic discus
Samuelson–Berkowitz algorithm
In mathematics, the Samuelson–Berkowitz algorithm efficiently computes the characteristic polynomial of an matrix whose entries may be elements of any unital commutative ring. Unlike the Faddeev–LeVer
Linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices. Linear algebra is central to almo
Fredholm alternative
In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algeb
3D projection
A 3D projection (or graphical projection) is a design technique used to display a three-dimensional (3D) object on a two-dimensional (2D) surface. These projections rely on visual perspective and aspe
Segre classification
The Segre classification is an algebraic classification of rank two symmetric tensors. The resulting types are then known as Segre types. It is most commonly applied to the energy–momentum tensor (or
Spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matri
Numerical range
In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex matrix A is the set where denotes the conjugate transpose of the vector . The numer
Kernel (linear algebra)
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map L
Spread of a matrix
In mathematics, and more specifically matrix theory, the spread of a matrix is the largest distance in the complex plane between any two eigenvalues of the matrix.
Matrix pencil
In linear algebra, if are complex matrices for some nonnegative integer , and (the zero matrix), then the matrix pencil of degree is the matrix-valued function defined on the complex numbers A particu
Orientation of a vector bundle
In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: E →B, an orientation of E means: for each fiber Ex,
Invariant subspace
In mathematics, an invariant subspace of a linear mapping T : V → V i.e. from some vector space V to itself, is a subspace W of V that is preserved by T; that is, T(W) ⊆ W.
Squeeze mapping
In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapp
Controlled invariant subspace
In control theory, a controlled invariant subspace of the state space representation of some system is a subspace such that, if the state of the system is initially in the subspace, it is possible to
Newton's identities
In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials
Bidiagonal matrix
In mathematics, a bidiagonal matrix is a banded matrix with non-zero entries along the main diagonal and either the diagonal above or the diagonal below. This means there are exactly two non-zero diag
Productive matrix
In linear algebra, a square nonnegative matrix of order is said to be productive, or to be a Leontief matrix, if there exists a nonnegative column matrix such as is a positive matrix.
Bunch–Nielsen–Sorensen formula
In mathematics, in particular linear algebra, the Bunch–Nielsen–Sorensen formula, named after James R. Bunch, Christopher P. Nielsen and Danny C. Sorensen, expresses the eigenvectors of the sum of a s
Linear independence
In the theory of vector spaces, a set of vectors is said to be linearly dependent if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination
CSS code
In quantum error correction, CSS codes, named after their inventors, Robert Calderbank, Peter Shor and Andrew Steane, are a special type of stabilizer code constructed from classical codes with some s
Dual basis in a field extension
In mathematics, the linear algebra concept of dual basis can be applied in the context of a finite extension L/K, by using the field trace. This requires the property that the field trace TrL/K provid
Matrix norm
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).
Coefficient matrix
In linear algebra, a coefficient matrix is a matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear equations.
Centrosymmetric matrix
In mathematics, especially in linear algebra and matrix theory, a centrosymmetric matrix is a matrix which is symmetric about its center. More precisely, an n×n matrix A = [Ai,j] is centrosymmetric wh
Translation of axes
In mathematics, a translation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x'y'-Cartesian coordinate system in which the x' axis is parallel to the x axis and k
Rayleigh's quotient in vibrations analysis
The Rayleigh's quotient represents a quick method to estimate the natural frequency of a multi-degree-of-freedom vibration system, in which the mass and the stiffness matrices are known. The eigenvalu
Corank
In mathematics, corank is complementary to the concept of the rank of a mathematical object, and may refer to the dimension of the left nullspace of a matrix, the dimension of the cokernel of a linear
Conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugate on each entry (the complex conj
Determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In
Matrix sign function
In mathematics, the matrix sign function is a matrix function on square matrices analogous to the complex sign function. It was introduced by J.D. Roberts in 1971 as a tool for model reduction and for
Faddeev–LeVerrier algorithm
In mathematics (linear algebra), the Faddeev–LeVerrier algorithm is a recursive method to calculate the coefficients of the characteristic polynomial of a square matrix, A, named after Dmitry Konstant
Generalized singular value decomposition
In linear algebra, the generalized singular value decomposition (GSVD) is the name of two different techniques based on the singular value decomposition (SVD). The two versions differ because one vers
Nonlinear eigenproblem
In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) eigenvalue problem to equations that depend nonlinearly on the eigenvalue. Speci
Quotient space (linear algebra)
In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N (read "V mod
Radial set
In mathematics, a subset of a linear space is radial at a given point if for every there exists a real such that for every Geometrically, this means is radial at if for every there is some (non-degene
Singular value decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis
Seminorm
In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Mi
Dual number
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form a + bε, where a and b are real numbers, and ε is a symbol taken to
Zero object (algebra)
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, and as a magma has a trivial structure,
Least-squares spectral analysis
Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the mos
Invariants of tensors
In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor are the coefficients of the characteristic polynomial , where is the
Tapering (mathematics)
In mathematics, physics, and theoretical computer graphics, tapering is a kind of shape deformation. Just as an affine transformation, such as scaling or shearing, is a first-order model of shape defo
Signal-flow graph
A signal-flow graph or signal-flowgraph (SFG), invented by Claude Shannon, but often called a Mason graph after Samuel Jefferson Mason who coined the term, is a specialized flow graph, a directed grap
Rotation of axes
In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes
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
Quasinorm
In linear algebra, functional analysis and related areas of mathematics, a quasinorm is similar to a norm in that it satisfies the norm axioms, except that the triangle inequality is replaced by for s
Zech's logarithm
Zech logarithms are used to implement addition in finite fields when elements are represented as powers of a generator . Zech logarithms are named after Julius Zech, and are also called Jacobi logarit
Matrix analysis
In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular topics out of many include; operations defined
Trace diagram
In mathematics, trace diagrams are a graphical means of performing computations in linear and multilinear algebra. They can be represented as (slightly modified) graphs in which some edges are labeled
Trilinear coordinates
In geometry, the trilinear coordinates x:y:z of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an exa
Antilinear map
In mathematics, a function between two complex vector spaces is said to be antilinear or conjugate-linear if hold for all vectors and every complex number where denotes the complex conjugate of Antili
Hamming space
In statistics and coding theory, a Hamming space (named after American mathematician Richard Hamming) is usually the set of all binary strings of length N. It is used in the theory of coding signals a
Jordan normal form
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finit
Eigengap
In linear algebra, the eigengap of a linear operator is the difference between two successive eigenvalues, where eigenvalues are sorted in ascending order. The Davis–Kahan theorem, named after Chandle
Transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix, often d
Relative dimension
In mathematics, specifically linear algebra and geometry, relative dimension is the dual notion to codimension. In linear algebra, given a quotient map , the difference dim V − dim Q is the relative d
Outline of linear algebra
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.
Flat (geometry)
In geometry, a flat or Euclidean subspace is a subset of a Euclidean space that is itself a Euclidean space (of lower dimension). The flats in two-dimensional space are points and lines, and the flats
Bra–ket notation
In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A
Barycentric coordinate system
In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points
Euclidean vector
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and directi
Fusion frame
In mathematics, a fusion frame of a vector space is a natural extension of a frame. It is an additive construct of several, potentially "overlapping" frames. The motivation for this concept comes from
Rota's basis conjecture
In linear algebra and matroid theory, Rota's basis conjecture is an unproven conjecture concerning rearrangements of bases, named after Gian-Carlo Rota. It states that, if X is either a vector space o
Fredholm's theorem
In mathematics, Fredholm's theorems are a set of celebrated results of Ivar Fredholm in the Fredholm theory of integral equations. There are several closely related theorems, which may be stated in te
Trace identity
In mathematics, a trace identity is any equation involving the trace of a matrix.
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 a
Indeterminate system
In mathematics, particularly in algebra, an indeterminate system is a system of simultaneous equations (e.g., linear equations) which has more than one solution (sometimes infinitely many solutions).
International Linear Algebra Society
The International Linear Algebra Society (ILAS) is a professional mathematical society organized to promote research and education in linear algebra, matrix theory and matrix computation. It serves th
Quadratic eigenvalue problem
In mathematics, the quadratic eigenvalue problem (QEP), is to find scalar eigenvalues , left eigenvectors and right eigenvectors such that where , with matrix coefficients and we require that , (so th
Choi's theorem on completely positive maps
In mathematics, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite-dimensional (matrix) C*-algebras. An infinite-dimensional algebraic gener
Entanglement-assisted stabilizer formalism
In the theory of quantum communication, the entanglement-assisted stabilizer formalism is a method for protecting quantum information with the help of entanglement shared between a sender and receiver
Norm (mathematics)
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obey
Commutation matrix
In mathematics, especially in linear algebra and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix into the vectorized form of its transpose. Specifically,
Eigenoperator
In mathematics, an eigenoperator, A, of a matrix H is a linear operator such that where is a corresponding scalar called an eigenvalue.
Hahn–Banach theorem
The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows
List of vector spaces in mathematics
This is a list of vector spaces in abstract mathematics, by Wikipedia page.
* Banach space
* Besov space
* Bochner space
* Dual space
* Euclidean space
* Fock space
* Fréchet space
* Hardy spa
Fangcheng (mathematics)
Fangcheng (sometimes written as fang-cheng or fang cheng) (Chinese: 方程; pinyin: fāng chéng) is the title of the eighth chapter of the Chinese mathematical classic Jiuzhang suanshu (The Nine Chapters o
Linear span
In mathematics, the linear span (also called the linear hull or just span) of a set S of vectors (from a vector space), denoted span(S), is defined as the set of all linear combinations of the vectors
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was publi
Modeshape
In applied mathematics, mode shapes are a manifestation of eigenvectors which describe the relative displacement of two or more elements in a mechanical system or wave front.A mode shape is a deflecti
Permanent (mathematics)
In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix. Bot
Three-dimensional rotation operator
This article derives the main properties of rotations in 3-dimensional space. The three Euler rotations are one way to bring a rigid body to any desired orientation by sequentially making rotations ab
Unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in (pronou
Delta operator
In mathematics, a delta operator is a shift-equivariant linear operator on the vector space of polynomials in a variable over a field that reduces degrees by one. To say that is shift-equivariant mean
Mixed linear complementarity problem
In mathematical optimization theory, the mixed linear complementarity problem, often abbreviated as MLCP or LMCP, is a generalization of the linear complementarity problem to include free variables.
Projectivization
In mathematics, projectivization is a procedure which associates with a non-zero vector space V a projective space , whose elements are one-dimensional subspaces of V. More generally, any subset S of
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,
Convex cone
In linear algebra, a cone—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, C is a cone
Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as sub
Sedrakyan's inequality
The following inequality is known as Sedrakyan's inequality, Bergström's inequality, Engel's form or Titu's lemma, respectively, referring to the article About the applications of one useful inequalit
Semi-simplicity
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is
Linear recurrence with constant coefficients
In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients (also known as a linear recurrence relation or linear difference equatio
Matrix calculus
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with re
Orthogonal Procrustes problem
The orthogonal Procrustes problem is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices and and asked to find an orthogonal matrix which most closely ma
Zero mode
In physics, a zero mode is an eigenvector with a vanishing eigenvalue. In various subfields of physics zero modes appear whenever a physical system possesses a certain symmetry. For example, normal mo
Orthogonalization
In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly independent set of vectors {v1, ... , vk}
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 th
Angles between flats
The concept of angles between lines in the plane and between pairs of two lines, two planes or a line and a plane in space can be generalized to arbitrary dimension. This generalization was first disc
Compressed sensing
Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and reconstructing a signal, by finding solu
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 easy example may be
Definite quadratic form
In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every non-zero vector of V. According to tha
Skew-Hermitian matrix
In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix is skew-He
Hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if
Eigenplane
In mathematics, an eigenplane is a two-dimensional invariant subspace in a given vector space. By analogy with the term eigenvector for a vector which, when operated on by a linear operator is another
Wild problem
In the mathematical areas of linear algebra and representation theory, a problem is wild if it contains the problem of classifying pairs of square matrices up to simultaneous similarity. Examples of w
Rank factorization
In mathematics, given a field , nonnegative integers , and a matrix , a rank decomposition or rank factorization of A is a factorization of A of the form A = CF, where and , where is the rank of .
Partial trace
In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued funct
Non-negative matrix factorization
Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usuall
Absolutely convex set
In mathematics, a subset C of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which
Homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, are a system of coordinates used in projective geom
Polarization identity
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If
Zassenhaus algorithm
In mathematics, the Zassenhaus algorithmis a method to calculate a basis for the intersection and sum of two subspaces of a vector space.It is named after Hans Zassenhaus, but no publication of this a
Row and column vectors
In linear algebra, a column vector is a column of entries, for example, which may also be viewed as an matrix for some .Similarly, a row vector is a row of entries,or equivalently a matrix for some .
Liouville space
In the mathematical physics of quantum mechanics, Liouville space, also known as line space, is the space of operators on Hilbert space. Liouville space is itself a Hilbert space under the Hilbert-Sch
Dimension (vector space)
In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. It is sometimes called Hamel dimension (after Georg Hamel) or al
Sesquilinear form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of
Orthonormal function system
An orthonormal function system (ONS) is an orthonormal basis in a vector space of functions. See basis (linear algebra), Fourier analysis, square-integrable, Hilbert space for more.
Maple (software)
Maple is a symbolic and numeric computing environment as well as a multi-paradigm programming language. It covers several areas of technical computing, such as symbolic mathematics, numerical analysis
Unitary transformation
In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the tran
K-frame
In linear algebra, a branch of mathematics, a k-frame is an ordered set of k linearly independent vectors in a vector space; thus k ≤ n, where n is the dimension of the space, and if k = n an n-frame
Standard basis
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as or ) is the set of vectors whose components are all zero, except one that equals
Function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, t
Self-adjoint
In mathematics, and more specifically in abstract algebra, an element x of a *-algebra is self-adjoint if . A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A col
Matrix congruence
In mathematics, two square matrices A and B over a field are called congruent if there exists an invertible matrix P over the same field such that PTAP = B where "T" denotes the matrix transpose. Matr
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 th
Weyl's inequality
In linear algebra, Weyl's inequality is a theorem about the changes to eigenvalues of an Hermitian matrix that is perturbed. It can be used to estimate the eigenvalues of a perturbed Hermitian matrix.
Scalar (mathematics)
A scalar is an element of a field which is used to define a vector space.In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vect
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 (e.g. a linear combination of x and y would be any e
Transpose of a linear map
In linear algebra, the transpose of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces. The transpose or algebraic
Bilinear form
In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements of which are called scalars). In other words, a
Immanant
In mathematics, the immanant of a matrix was defined by Dudley E. Littlewood and Archibald Read Richardson as a generalisation of the concepts of determinant and permanent. Let be a partition of an in
Linear form
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers
Balanced set
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field with an absolute value function ) is a set such that for all scalars satisfying T
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 addition for ma
Golden–Thompson inequality
In physics and mathematics, the Golden–Thompson inequality is a trace inequality between exponentials of symmetric and Hermitian matrices proved independently by and . It has been developed in the con
Sublinear function
In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space is a real-valued function w
Basis function
In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as e
Linear inequality
In mathematics a linear inequality is an inequality which involves a linear function. A linear inequality contains one of the symbols of inequality:. It shows the data which is not equal in graph form
Z-order curve
In mathematical analysis and computer science, functions which are Z-order, Lebesgue curve, Morton space-filling curve, Morton order or Morton code map multidimensional data to one dimension while pre
Schur product theorem
In mathematics, particularly in linear algebra, the Schur product theorem states that the Hadamard product of two positive definite matrices is also a positive definite matrix.The result is named afte
Normal basis
In mathematics, specifically the algebraic theory of fields, a normal basis is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois gr
Stokes operator
The Stokes operator, named after George Gabriel Stokes, is an unbounded linear operator used in the theory of partial differential equations, specifically in the fields of fluid dynamics and electroma
Invertible matrix
In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate), if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix a
Spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a