Matrix Structure & Linear Maps

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Abstractions about matrices and linear transformations classified by algebraic form, spectral behavior, sign constraints, decomposition, and canonical representation. They include structured matrix families, quadratic and sesquilinear forms, rank defects, products, and computational operations.

48 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.
• 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.
• Dimension (vector space) — The cardinality of any basis of a vector space over a specified field, well-defined because all bases have the same cardinality.
• 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.
• Frobenius normal form — Replace a square matrix over a field by the unique block diagonal matrix of companion matrices determined by its divisibility-ordered invariant factors, thereby deciding similarity without splitting the characteristic polynomial.
• 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.
• Linear map — A function between vector spaces that preserves vector addition and scalar multiplication, equivalently preserving every finite linear combination.
• Linear matrix inequality — A convex constraint requiring an affine combination of symmetric or Hermitian matrices to be positive semidefinite.
• 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.
• Orthostochastic matrix — A doubly stochastic matrix obtained by squaring the entries of a real orthogonal matrix componentwise.
• Outer product — Map two coordinate vectors to the rank-at-most-one matrix whose ij entry is the product of the first vector’s i component and the second vector’s j component.
• 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.
• Rational dependence — The property that a finite collection of numbers satisfies a nontrivial linear relation with rational coefficients.
• 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.
• Spread of a matrix — The maximum complex-plane distance between any two eigenvalues of a square matrix.
• 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.
• Weyr canonical form — A canonical matrix form obtained by regrouping Jordan chains by level, yielding a block structure particularly suited to describing matrices that commute with a given operator.
• Z-matrix (mathematics) — A real square matrix whose every off-diagonal entry is nonpositive.