This page gathers the 95 dictionary entries belonging to the domain “Linear algebra”, as labelled by Wiktionary. Each word leads to its full entry: definitions, etymology, pronunciation, examples.
- adjoint noun Transpose conjugate.
- affine transformation noun A geometric transformation that preserves lines and parallelism, but in general not lengths or angles; (more formally) an automorphism of…
- antisymmetric adj Whose transpose equals its negative (i.e., Mᵀ = −M);
- augmented matrix noun Matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on…
- back substitution noun A method of solving linear systems that have been transformed into row echelon form.
- basis noun In a vector space, a linearly independent set of vectors spanning the whole vector space.
- bilinear adj Linear (preserving linear combinations) in each variable.
- bilinear form noun A function of two arguments from the same vector space which maps onto a field of scalars, which acts like a linear form with respect to…
- bra noun One of the two vectors in the standard notation for describing quantum states in quantum mechanics, the row vector; the other (column)…
- characteristic polynomial noun The polynomial produced from a given square matrix by first subtracting the appropriate identity matrix multiplied by an indeterminant and…
- coefficient matrix noun A matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear…
- cofactor noun The signed determinant of the submatrix produced by removing the row and column containing a specified element; primarily used in the…
- congruence noun Matrix similarity by an orthogonal matrix.
- contravariance noun Of vectors, the property of scaling inversely with a change of basis, as opposed to covariance.
- contravariant adj Scaling inversely with a change of basis.
- convex combination noun A linear combination (of vectors in Euclidean space) in which the coefficients are non-negative and all add up to one.
- Cramer's rule noun An explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a…
- cross term noun In a quadratic form, a term which is not a squared variable; i.e., a term with more than one variable.
- defective matrix noun A square (n×n) matrix that has fewer than n linearly independent eigenvectors, and is therefore not diagonalisable.
- determinant noun A scalar that encodes certain characteristics of a given transformation matrix; the unique scalar function over square matrices which is…
- diagonal element noun An element on the main diagonal of a square matrix; that is, an element in row k and column k, where k is an integer between 1 and the…
- dimension noun The number of elements of any basis of a vector space.
- direct sum noun A linear sum in which the intersection of the summands has dimension zero. Equivalently, a linear sum of two subspaces, any vector of which…
- dual adj Being the space of all linear functionals of (some other space).
- echelon adj Of a matrix: having undergone Gaussian elimination with the result that the leading coefficient or pivot (that is, the first nonzero number…
- echelon form noun row echelon form
- eigen- prefix Forms terms pertaining to or related to eigenvectors, eigenvalues; especially for naming mathematical objects which are not affected by a…
- eigendecomposition noun The factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors.
- eigenpair noun The mathematical pair of an eigenvector (ẕ), and its associated eigenvalue (λ). This is written as (λ,ẕ)
- eigenspace noun The linear subspace consisting of all eigenvectors associated with a particular eigenvalue, together with the zero vector.
- eigenvalue noun A scalar λ, such that there exists a non-zero vector x (a corresponding eigenvector) for which the image of x under a given linear…
- eigenvector noun A vector that is only scaled (not rotated out of its span) under a particular linear transformation; a left or right eigenvector depending…
- elementary matrix noun A square matrix which is obtained from the identity matrix by a single elementary row (or column) operation.
- entry noun A term at any position in a matrix.
- flag noun A sequence of subspaces of a vector space, beginning with the null space and ending with the vector space itself, such that each member of…
- full rank adj Having the highest possible rank for its size; having a rank (dimension) equal to the number of columns or the number of rows.
- Hermitian matrix noun A square matrix A with complex entries that is equal to its own conjugate transpose, i.e., such that A=A^†.
- homogeneous adj Such that all the constant terms are zero.
- immanant noun A function or property of a matrix, defined as a generalization of the concepts of determinant and permanent.
- inner product noun A generalization of the dot product for vectors of any dimensionality that may or may not be complex-numbered.
- inverse matrix noun Of a matrix A, another matrix B such that A multiplied by B and B multiplied by A both equal the identity matrix.
- invertible matrix noun Any n×n square matrix for which there exists a corresponding inverse matrix (i.e., a second (or possibly the same) matrix such that when…
- kernel noun The set of elements of M which are mapped to the additive identity 0 of N.
- ket noun A column vector, in Hilbert space, especially as representing the state of a quantum mechanical system; the complex conjugate transpose of…
- left eigenvalue noun An eigenvalue corresponding to the left transform of a given linear operator; i.e. λ such that urm A=uλ.
- left eigenvector noun An eigenvector corresponding to the left transform of a given linear operator; i.e. u such that urm A=uλ.
- left nullspace noun The vector space of all row vectors whose dot products with the columns of a given matrix are zero; the nullspace of the transpose of a…
- linear combination noun a sum, each of whose summands is an appropriate vector times an appropriate scalar (or ring element)
- linear span noun A linear subspace generated by all linear combinations of a given subset of vectors of a given vector space.
- linear subspace noun A subset of vectors of a vector space which is closed under the addition and scalar multiplication of that vector space.
- linear transformation noun A map between vector spaces which preserves the operations of vector addition and scalar multiplication.
- lu noun A matrix decomposition which writes a matrix as the product of a lower and upper triangular matrix, used in numerical analysis to solve…
- minimal polynomial noun For a given square matrix M over some field K, the smallest-degree monic polynomial over K which, when applied to M, yields the zero matrix.
- multilinear form noun Given a vector space V over a field K of scalars, a mapping Vᵏ → K that is linear in each of its arguments;
- nonsingular adj Invertible.
- nontrivial solution noun Any other answers apart from ⃑x=⃑0 to the linear system A⃑x=⃑0, signifying linear dependence of the linear system.
- normal adj Which commutes with its conjugate transpose.
- orthogonal matrix noun A square matrix whose columns, considered as vectors, are orthonormal to each other. This implies that the transpose of such a matrix is…
- orthonormal adj Of a set of vectors, both orthogonal and normalized.
- overdetermined adj Having more equations than variables.
- permanent noun Given an n×n matrix a_ij,, the sum over all permutations π, of ∏ᵢ₌₁ⁿa_iπ(i).
- polar cone noun Relative to a set S in Rⁿ, the set of points y such that the inner product of x and y is not positive for any point x in S.
- positive-definite adj Said of a real, square matrix: that the product of it with a column vector on its right side and the transpose of that column vector on its…
- projection noun An idempotent linear transformation which maps vectors from a vector space onto a subspace.
- projection matrix noun A transformation matrix whose associated transformation is a projection.
- QR decomposition noun A decomposition of a matrix into a product of a unitary matrix (orthogonal matrix) and an upper-triangular matrix.
- rank noun The maximal number of linearly independent columns (or rows) of a matrix.
- rank-deficient adj Not having full rank; having a rank (dimension) less than the number of columns and the number of rows.
- right eigenvalue noun An eigenvalue corresponding to the right transform of a given linear operator; i.e. λ such that rm Au=λu.
- right eigenvector noun An eigenvector corresponding to the right transform of a given linear operator; i.e. u such that rm Au=λu.
- row echelon form noun The stepped appearance of a matrix that has undergone Gaussian elimination with the result that the leading coefficient or pivot (that is…
- row-equivalence noun A relation between two matrices of the same size, such that every row of one matrix is a linear combination of the rows of the other…
- row-reduced adj Said of a matrix which has undergone row reduction; i.e., of a matrix which has a 1 as the leading entry of each one of its non-zero rows…
- scalar field noun The field (algebraic structure) for which scalar multiplication is defined for a given vector space; field of scalars.
- scalar multiplication noun Multiplication of a vector by a scalar (which belongs to the vector space's field of scalars).
- scalar product noun The product of two vectors computed as the sum of the corresponding elements of the vectors, or, equivalently, as the product of the…
- Schur decomposition noun A matrix decomposition allowing one to write complex square matrices as similar to a triangular matrix whose diagonal elements are the…
- self-dual adj Of a vector space, isomorphic to its dual space.
- semisimple adj For which every invariant subspace has an invariant complement, equivalent to the minimal polynomial being squarefree.
- similar adj Of two square matrices; being such that a conjugation sends one matrix to the other.
- similarity noun The property of two matrices being similar.
- singular adj Having no inverse.
- singular value decomposition noun A particular type of factorisation of a matrix into a product of three matrices, of which the second is a diagonal matrix that has as the…
- skew-symmetric adj Of a matrix, satisfying A^( textsf )T=-A, i.e. having entries on one side of the diagonal that are the additive inverses of their…
- spectrum noun The set of eigenvalues of a matrix.
- symplectic group noun For given field F and positive integer n, the group of 2n×2n symplectic matrices with elements in F.
- symplectic matrix noun For given field F (especially the real numbers), even order 2n and nonsingular skew-symmetric matrix Ω, any 2n×2n matrix M with elements in…
- syzygy noun A relation between generators of a module.
- tensor noun A mathematical object that describes linear relations on scalars, vectors, matrices and other algebraic objects, and is represented as a…
- trace noun The sum of the diagonal elements of a square matrix.
- transpose adj Created by transposing a specified matrix.
- trilinear adj Linear (preserving linear combinations) in each variable.
- unitary adj Whose inverse is equal to its adjoint.
- vector product noun a vector with the size given by the product of two vectors computed as the product of the magnitudes of the vectors and the sine of the…
- zero vector noun A vector 0 in a vector space V such that for any v∈V, 0+v=v; a vector whose value in every dimension is zero; i.e., ⃑0=<0,0,…,0>.