List of matrices

From Free net encyclopedia

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
• 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 H_ij = (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.
• 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 M^q = 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 = A^T.
• 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., a_ij = a_{n−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.
• 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, A^T = −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 = A^T.
• 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.