239 articles
Invertible matrix
An invertible matrix, also called a non-singular matrix, is a square matrix with an inverse that multiplies it to give the identity; such an inverse exists exactly when the determinant is nonzero.
Jacobi method
The Jacobi method, also known as Jacobi iteration, is an iterative algorithm of numerical linear algebra that solves systems of linear equations; Jacobi described it in 1845.
Jacobi's formula
Jacobi's formula is a theorem of matrix calculus stating that the differential of a square matrix's determinant equals the trace of its adjugate times the matrix differential.
Jacobian matrix and determinant
The Jacobian matrix of a vector-valued function is the matrix of its first-order partial derivatives; when square, its Jacobian determinant measures local volume scaling and appears in change of variables.
Jordan block
A Jordan block is a square matrix with a single number λ on its main diagonal, 1s on the diagonal above, and 0s everywhere else.
Jordan normal form
The Jordan normal form, also called the Jordan canonical form, is a block-diagonal matrix representing a linear operator when a matrix cannot be diagonalized, with eigenvalues on the diagonal and 1s on the superdiagonal.
Kaczmarz method
The Kaczmarz method, also called the Kaczmarz algorithm, is an iterative method for solving systems of linear equations by projecting onto one equation's hyperplane at a time, widely used in computed tomography as ART.
Kahan summation algorithm
The Kahan summation algorithm, also called compensated summation, is a method for adding floating-point numbers that captures each addition's rounding error in a compensation variable, cutting accumulated error; William Kahan published it in 1965.
Kernel (linear algebra)
The kernel of a linear map, also called its null space, is the set of all domain vectors the map sends to zero; it is a linear subspace of the domain.
Khatri-Rao product
The Khatri-Rao product, or column-wise Kronecker product, is a matrix product that applies the Kronecker product column by column, introduced by Khatri and Rao in 1968.
Kronecker delta
The Kronecker delta, written δij, is a mathematical symbol that equals 1 when its two indices are equal and 0 otherwise, serving as the index-notation form of the identity matrix.
Kronecker product
The Kronecker product, also called the tensor product of matrices, replaces each entry of a matrix A with A's entry times matrix B, yielding a block matrix.
Krylov subspace
A Krylov subspace is the vector space spanned by b, Ab, A²b, and so on for a matrix A, named for Aleksey Krylov in 1931.
Krylov subspace methods
Krylov subspace methods are iterative projection methods for large linear systems, eigenvalue problems, and matrix functions, building approximations from repeated matrix-vector products; examples include CG, GMRES, and BiCGSTAB.
Lanczos algorithm
The Lanczos algorithm, also called the Lanczos method, is an iterative method introduced by Cornelius Lanczos in 1950 for finding extreme eigenvalues of large sparse symmetric matrices.
Landweber iteration
Landweber iteration is a gradient-descent method for solving ill-posed linear inverse problems, introduced by Louis Landweber in 1951 and used in image deblurring and tomography.
LAPACK
LAPACK (Linear Algebra Package) is a freely available Fortran library, first released in 1992, for solving linear systems, least-squares, eigenvalue, and singular value problems for dense and banded matrices.
Leibniz formula for determinants
The Leibniz formula for determinants expresses the determinant of an n × n matrix as a signed sum over all n! permutations, a definition introduced by Leibniz in 1693.
Levi-Civita symbol
The Levi-Civita symbol, also called the permutation symbol, is a collection of numbers based on the sign of a permutation, equal to +1, −1, or 0.
Linear algebra
Linear algebra is the branch of mathematics that studies vector spaces and linear maps between them, together with the matrices, determinants, and eigenvalues used to compute with them.
Linear combination
A linear combination is an expression of the form c1v1 + ... + ckvk, formed by scaling vectors and adding them; it underlies span, independence, and bases.
Linear function
A linear function, also called a linear map or linear transformation, is a function between vector spaces that preserves vector addition and scalar multiplication; it is represented by a matrix.
Linear independence
Linear independence is the property of a set of vectors where no vector is a linear combination of the others; it is half of the definition of a basis.
Linear map
A linear map, also called a linear transformation, is a function between vector spaces that preserves addition and scalar multiplication, exemplified by rotations and projections.
Linear span
The linear span, also called the linear hull, of a set of vectors is the set of all their finite linear combinations, and it forms the smallest subspace containing them.
Linear subspace
A linear subspace, also called a vector subspace, is a subset of a vector space that is itself a vector space under the same addition and scalar multiplication; examples include lines and planes through the origin.
Linearity
Linearity is the property of a function or system satisfying additivity and homogeneity, meaning T(u+v) = T(u)+T(v) and T(av) = aT(v); it enables superposition, unlike nonlinear problems.
Low-rank approximation
A low-rank approximation replaces a matrix with one of much lower rank that preserves its dominant structure, used for compression, denoising, and fast linear algebra on large data matrices.
LU decomposition
The LU decomposition, also called LU factorization or LR decomposition, expresses a matrix as the product of a lower triangular matrix L and an upper triangular matrix U.
M-matrix
An M-matrix is a real square matrix with nonpositive off-diagonal entries and nonnegative principal minors, named for Minkowski and used in numerical analysis, economics, and probability.