List of named matrices
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Listed below are some important classes of matrices used in mathematics:
- (0,1)-matrix or binary matrix — a matrix with all elements either 0 or 1.
- Adjacency matrix — a square matrix used to represent the connectivity of a graph.
- Alternating sign matrix — a generalization of permutation matrices that arises from Dodgson condensation.
- Anti-diagonal matrix — a square matrix with all entries off the anti-diagonal equal to zero.
- Anti-Hermitian matrix — another name for a skew-Hermitian matrix.
- Anti-symmetric matrix — another name for a skew-symmetric matrix.
- Band matrix — a square matrix with all entries off a diagonally bordered "band" equal to zero.
- Bézout matrix — a tool for efficient location of polynomial zeros
- Block diagonal matrix — a block matrix with entries only on the diagonal.
- Block matrix — a matrix partitioned in sub-matrices called blocks.
- Cartan matrix
- Centrosymmetric matrix — a matrix symmetric about its center; i.e., aij = an−i+1,n−j+1
- Circulant matrix — a matrix where each row is a circular shift of its predecessor.
- Companion matrix — the companion matrix of a polynomial is a special form of matrix, whose eigenvalues are equal to the roots of the polynomial.
- Coxeter matrix
- Diagonal matrix — a square matrix with all entries off the main diagonal equal to zero.
- Diagonalizable matrix — a square matrix similar to a diagonal matrix. It has a complete set of linearly independent eigenvectors.
- Distance matrix — a square matrix containing the distances, taken pairwise, of a set of points.
- Gell-Mann matrices
- Generalized permutation matrix — a square matrix with precisely one nonzero element in each row and column.
- Gramian matrix — a real symmetric matrix that can be used to test for linear independence of any function.
- Hadamard matrix — square matrix with entries +1, −1 whose rows are mutually orthogonal.
- Hamiltonian matrix — Used in linear quadratic regulator (LQR) systems.
- Hankel matrix — a matrix with constant off diagonals; also an upside down Toeplitz matrix. A square Hankel matrix is symmetric.
- Hermitian matrix — a square matrix which is equal to its conjugate transpose, A = A*.
- Hessenberg matrix — an "almost" triangular matrix, for example, an upper Hessenberg matrix has zero entries below the first subdiagonal.
- Hessian matrix — a square matrix of second partial derivatives of a scalar-valued function.
- Hilbert matrix — a Hankel matrix with elements Hij = (i + j − 1)−1.
- Householder matrix — a transformation matrix widely used in matrix algorithms.
- Idempotent matrix — a matrix that has the property A2 = A.
- Identity matrix — a square diagonal matrix, with all entries on the main diagonal equal to 1, and the rest 0.
- Invertible matrix — a square matrix with a multiplicative inverse.
- Jacobian matrix —
- Logical matrix — a k-dimensional array of boolean values that represents a k-adic relation.
- Matrix exponential — defined by the exponential series
- Matrix representation of conic sections
- Nilpotent matrix — a square matrix M such that Mq = 0 for some positive integer q.
- Nonnegative matrix — a matrix with all nonnegative entries.
- Normal matrix — a square matrix that commutes with its conjugate transpose. Normal matrices are precisely the matrices to which the spectral theorem applies.
- Orthogonal matrix — a matrix whose inverse is equal to its transpose, A−1 = AT.
- Orthonormal matrix — matrix whose columns are orthonormal vectors.
- Overlap matrix
- Pascal matrix — matrix containing the entries of Pascal's triangle.
- Pauli matrices
- Payoff matrix
- Pentadiagonal matrix — a matrix with the only nonzero entries on the main diagonal and the two diagonals just above and below the main one.
- Permutation matrix — matrix representation of a permutation.
- Persymmetric matrix — a matrix that is symmetric about its northeast-southwest diagonal, i.e., aij = an−j+1,n−i+1
- Pick matrix — occurs in the study of analytical interpolation problems
- Positive-definite matrix — a Hermitian matrix with every eigenvalue positive.
- Positive matrix — a matrix with all positive entries.
- Random matrix — a matrix of given type and size whose entries consist of random numbers from some specified distribution.
- Row echelon form - used in Gaussian elimination
- S matrix — in physics
- Singular matrix — a noninvertible square matrix.
- Similarity matrix — a matrix of scores which express the similarity between two data points.
- Skew-Hermitian matrix — a square matrix which is equal to the negative of its conjugate transpose, A* = −A.
- Skew-symmetric matrix — a matrix which is equal to the negative of its transpose, AT = −A.
- Sparse matrix — containing mostly zeros
- Square matrix — an n by n matrix. The set of all square matrices form an associative algebra with identity.
- Stochastic matrix — a positive matrix describing a stochastic process. The sum of entries of any row is one.
- Substitution matrix
- Sylvester matrix — a square matrix whose entries come from coefficients of two polynomials. The sylvester matrix is nonsingular if and only if the two polynomials are co-prime to each other.
- Symmetric matrix — a square matrix which is equal to its transpose, A = AT.
- Symplectic matrix — a square matrix preserving a standard skew-symmetric form.
- Toeplitz matrix — a matrix with constant diagonals.
- Totally positive matrix — a matrix with determinants of all its square submatrices positive. It is used in generating the reference points of Bézier curve in computer graphics.
- Totally unimodular matrix — a matrix for which every non-singular square submatrix is unimodular. This has some implications in the linear programming relaxation of an integer program.
- Transformation matrix
- Transition matrix — a matrix representing the probabilities of changing from one state to another
- Triangular matrix — a matrix with all entries above the main diagonal equal to zero (lower triangular) or with all entries below the main diagonal equal to zero (upper triangular).
- Tridiagonal matrix — a matrix with the only nonzero entries on the main diagonal and the diagonals just above and below the main one.
- Unimodular matrix — a square matrix with determinant +1 or −1.
- Unitary matrix — a square matrix whose inverse is equal to its conjugate transpose, A−1 = A*.
- Vandermonde matrix — a row consists of 1, a, a2, a3, etc., and each row uses a different variable
- Walsh matrix — a square matrix, with dimensions a power of 2, the entries of which are +1 or -1.
- Wronskian
- Zero matrix — a matrix with all entries equal to zero.