239 articles
Exterior power
The k-th exterior power ΛᵏV of a vector space or module V is built from the k-fold tensor power by forcing tensors with a repeated factor to vanish.
Factor analysis
Factor analysis is a statistical method that explains correlations among observed variables through fewer unobserved factors; invented by Charles Spearman in 1904, it is a core psychometrics tool.
Frobenius norm
The Frobenius norm, also called the Hilbert-Schmidt norm, is a matrix norm defined as the square root of the sum of squared absolute entries, named for Ferdinand Georg Frobenius.
Gauss–Newton algorithm
The Gauss–Newton algorithm, or Gauss–Newton method, is an iterative method for minimizing nonlinear least squares problems, solving a linear least squares step each iteration without second derivatives.
Gauss–Seidel method
The Gauss–Seidel method is an iterative method for solving linear systems, a refinement of the Jacobi method described by Ludwig Seidel in 1874 and named for him and Carl Friedrich Gauss.
Gaussian elimination
Gaussian elimination, also known as row reduction, is an algorithm that solves systems of linear equations by reducing a matrix to upper triangular form and applying back substitution.
Gaussian orthogonal ensemble
The Gaussian orthogonal ensemble (GOE) is a probability distribution on real symmetric matrices whose law is invariant under conjugation by orthogonal matrices, introduced by Eugene Wigner for nuclear level statistics.
Gaussian symplectic ensemble
The Gaussian symplectic ensemble (GSE) is a random matrix ensemble introduced by Freeman Dyson in 1962 for quantum systems with time-reversal invariance but no rotational symmetry.
General linear group
The general linear group GL_n(R) is the group of all invertible n by n matrices over a ring R, forming a group under matrix multiplication.
Generalized eigenvector
A generalized eigenvector of a matrix A is a nonzero vector v with (A − λI)^p v = 0 for some p; ordinary eigenvectors are the case p = 1.
Generalized minimal residual method
The generalized minimal residual method (GMRES) is an iterative Krylov subspace method for solving large sparse nonsymmetric linear systems, proposed by Saad and Schultz in 1986.
Gershgorin circle theorem
The Gershgorin circle theorem is a result in matrix analysis stating that every eigenvalue of a square matrix lies in disks centered at the diagonal entries, published in 1931.
Givens rotation
A Givens rotation is a plane rotation in numerical linear algebra, acting on two coordinates of a vector to turn one chosen entry into zero.
GNU Octave
GNU Octave is a free, high-level programming language and interactive environment for numerical computations, with syntax mostly compatible with Matlab. It runs on Linux, macOS, and Windows.
Graded vector space
A graded vector space, also called a graded linear space, is a vector space decomposed as a direct sum of subspaces indexed by degree, often integers.
Gram matrix
The Gram matrix is a square matrix whose entries are the pairwise inner products of a set of vectors, encoding lengths, angles, and linear dependence; it is always positive semidefinite.
GraphBLAS
GraphBLAS (Graph Basic Linear Algebra Subprograms) is a specification for an API that expresses graph algorithms as sparse matrix operations over semirings, developed by the open GraphBLAS Forum and modeled on the BLAS standard.
Hadamard product (matrices)
The Hadamard product, also called the Schur product or entrywise product, multiplies two matrices of the same size entry by entry, unlike ordinary matrix multiplication.
Hankel matrix
A Hankel matrix is a square or rectangular matrix whose entries are constant along each anti-diagonal, named after the German mathematician Hermann Hankel.
Hermitian matrix
A Hermitian matrix, also called a self-adjoint matrix, is a complex square matrix that equals its own conjugate transpose; its eigenvalues are always real, and it is unitarily diagonalizable.
Hessian matrix
The Hessian matrix, or Hesse matrix, is the square matrix of a function's second-order partial derivatives, used to classify critical points as minima, maxima, or saddle points.
Higher-order singular value decomposition
The higher-order singular value decomposition (HOSVD), also known as higher-order SVD, is a multilinear generalization of the matrix SVD, introduced by De Lathauwer, De Moor, and Vandewalle in 2000.
Hilbert matrix
The Hilbert matrix is the square matrix with entries 1/(i + j − 1), the standard example of a badly ill-conditioned matrix in numerical linear algebra.
Homogeneous function
A homogeneous function, also known as a homogeneous map, scales by a fixed power when all its arguments are scaled by the same factor, as in the Cobb–Douglas function.
Householder transformation
The Householder transformation, also called a Householder reflection, is a symmetric orthogonal matrix reflecting vectors across a hyperplane, widely used in QR factorization since Alston Householder introduced it in 1958.
Identity matrix
The identity matrix, also called the unit matrix, is the square matrix with ones on the diagonal and zeros elsewhere that leaves any matrix unchanged under multiplication.
Incomplete LU factorization
An incomplete LU factorization (ILU) approximates a sparse matrix as LU plus a discarded remainder R, keeping factors sparse and serving mainly as a preconditioner for Krylov methods like GMRES.
Invariant subspace
An invariant subspace is a subspace of a vector space that a linear operator maps into itself, the concept underlying eigenvectors, the Jordan canonical form, and a famous open problem.
Inverse eigenvalue problem
The inverse eigenvalue problem asks whether a matrix with prescribed eigenvalues exists within a given structural class, such as symmetric or nonnegative matrices, and how to construct it.
Inverse iteration
Inverse iteration is an eigenvalue algorithm that finds the eigenvector whose eigenvalue lies nearest a chosen shift, working by the power method applied to the inverse shifted matrix.