Structured Matrices & Linear Maps

← Back to Domain-Specific Families

Abstractions about matrices defined by special entry patterns or algebraic constraints — banded, circulant, Hadamard, Hankel, and Pascal forms, doubly stochastic matrices — alongside factorizations like LU and Schur complement, and related linear maps, bilinear forms, and scalar operations.

43 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• 3D rotation group — The Lie group SO(3) of orientation-preserving linear isometries of three-dimensional Euclidean space, represented by orthogonal matrices of determinant one.
• Arrowhead matrix — A square matrix whose only potentially nonzero entries lie on the main diagonal and one selected row and matching column.
• Balancing domain decomposition method — A nonoverlapping domain-decomposition preconditioner that combines independent subdomain solves with a coarse correction assembled from local nullspaces to solve symmetric positive-definite finite-element systems.
• Bidiagonal matrix — A banded matrix whose potentially nonzero entries lie only on the main diagonal and one adjacent superdiagonal or subdiagonal, yielding simple determinants, eigenvalues, products, and efficient structured computations.
• Block LU decomposition — Factorization of a partitioned matrix into lower and upper block-triangular factors, governed by Schur complements.
• Bohemian matrices — A family of matrices whose entries are restricted to a fixed finite discrete population, often bounded-height integers, sometimes with additional Toeplitz, Hessenberg or other structure.
• Butson-type Hadamard matrix — A complex Hadamard matrix whose entries are q-th roots of unity.
• Cauchy matrix — A structured matrix with entries 1/(x_i−y_j) for distinct parameter sequences with nonzero cross-differences, possessing explicit determinant, inverse, displacement rank, and totally nonsingular submatrix formulas.
• Circulant matrix — A square matrix generated by cyclically shifting one row, equivalently with entries depending only on index difference modulo n, and diagonalized by the discrete Fourier transform.
• Complex Hadamard matrix — A square complex matrix whose entries all have unit modulus and whose rows are mutually orthogonal.
• Computational complexity of matrix multiplication — The asymptotic resources required to multiply matrices, summarized algebraically by the smallest feasible exponent and operationally by finite-size costs and stability.
• Corank — A rank-deficiency quantity, commonly the codomain dimension minus rank of a linear map or matrix, equivalently the dimension of its cokernel in finite-dimensional linear algebra.
• Crout matrix decomposition — An LU factorization convention placing arbitrary diagonal entries in the lower triangular factor and unit diagonal entries in the upper factor, with pivoting when needed.
• Defective matrix — A square matrix lacking a full basis of eigenvectors and therefore not diagonalizable over the stated field.
• Definite quadratic form — A real quadratic form that is strictly positive on every nonzero vector or strictly negative on every nonzero vector.
• Doubly stochastic matrix — A square nonnegative matrix whose every row and column sums to one, equivalently a convex combination of permutation matrices and a point in the Birkhoff polytope.
• Exchange matrix — The permutation matrix with ones on the antidiagonal that reverses coordinate, row, or column order.
• Hadamard product (matrices) — The entrywise product of two matrices of identical shape, multiplying corresponding entries without summing across indices.
• Hankel matrix — A matrix whose entries are constant along every anti-diagonal, so each entry depends only on the sum of its row and column indices.
• Linear complex structure — A real-linear endomorphism J of a real vector space satisfying J squared equals minus the identity, thereby defining multiplication by complex scalars.
• Linear least squares — Approximation of an overdetermined or rank-deficient linear system by choosing parameters that minimize a quadratic residual norm.
• M-matrix — A real Z-matrix expressible as a nonnegative scalar multiple of the identity minus a nonnegative matrix with scalar at least its spectral radius.
• Matrix congruence — An equivalence relation on square matrices in which B equals transpose-P times A times P for an invertible change-of-basis matrix P.
• Metzler matrix — A real matrix whose off-diagonal entries are all nonnegative, serving as the continuous-time generator form for positive linear systems.
• Minor (linear algebra) — The determinant of a square submatrix obtained by selecting equal-size subsets of a matrix’s rows and columns.
• Modal matrix — A matrix whose columns are eigenvectors of a square matrix, used as the change of basis that diagonalizes it when a full eigenbasis exists.
• Monotone matrix — A real square matrix A for which componentwise nonnegativity of Ax implies componentwise nonnegativity of x, equivalently an invertible matrix with a nonnegative inverse.
• Mutual coherence (linear algebra) — The largest absolute normalized inner product between distinct columns of a matrix or atoms of a dictionary.
• Orthogonal Procrustes problem — A least-squares alignment problem seeking the orthogonal transformation that best maps one matrix configuration to another.
• Orthostochastic matrix — A doubly stochastic matrix obtained by squaring the entries of a real orthogonal matrix componentwise.
• Pascal matrix — A lower-triangular, upper-triangular or symmetric matrix whose entries are binomial coefficients arranged according to Pascal’s triangle.
• Quaternionic eigenvalue problem — The problem of finding left or right eigenvalues and eigenvectors of a matrix with quaternion entries, where noncommutativity makes the side of scalar multiplication constitutive.
• Quincunx matrix — The two-by-two integer matrix with rows (1, -1) and (1, 1), generating the diagonal same-parity square sublattice.
• Scalar multiplication — The vector-space or module operation that combines a scalar with a vector to produce another vector.
• Schur complement method — A nonoverlapping domain-decomposition method eliminating subdomain interiors and solving the remaining interface Schur-complement system.
• Semilinear map — An additive map between vector spaces whose scalar multiplication is respected after applying a fixed field automorphism.
• Sesquilinear form — A two-argument form on complex vector spaces that is linear in one argument and conjugate-linear in the other.
• Transfer matrix — The block-Toeplitz linear operator induced by a refinement mask whose eigenstructure characterizes refinable functions and their regularity.
• Transpose of a linear map — The induced linear map between dual spaces obtained by precomposing functionals with the original map.
• Transpositions matrix — A power-of-two square matrix generated from one vector by indexing entries with the bitwise XOR of row and column indices, so every row and column is a permutation of the vector.
• Unimodular matrix — A square integer matrix with determinant plus or minus one, equivalently an integer matrix invertible over the integers.
• Weakly chained diagonally dominant matrix — A weakly diagonally dominant matrix in which every non-strict row can reach a strictly dominant row through a directed chain of nonzero off-diagonal entries.
• Z-matrix (mathematics) — A real square matrix whose every off-diagonal entry is nonpositive.