List of matrices

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This page lists some important classes of matrices used in mathematics, science and engineering:

Contents 1 Matrices in mathematics 2 Constant matrices 3 Matrices used in statistics 4 Matrices used in graph theory 5 Matrices used in science and engineering 6 Other matrix-related terms and definitions

[edit] Matrices in mathematics

• (0,1)-matrix — a matrix with all elements either 0 or 1. Also called a binary matrix.
• 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.
• Augmented matrix — a matrix whose rows are concatenations of the rows of two smaller matrices.
• Band matrix — a square matrix whose non-zero entries are confined to a diagonal band.
• Bézout matrix — a square matrix which may be used as a tool for the efficient location of polynomial zeros
• Bidiagonal matrix — a matrix with elements only on the main diagonal and either the superdiagonal or subdiagonal (sometimes defined differently - see article).
• Binary matrix — another name for a (0,1)-matrix (a matrix whose coefficients are all either 0 or 1).
• Block-diagonal matrix — a block matrix with entries only on the diagonal.
• Block matrix — a matrix partitioned in sub-matrices called blocks.
• Block tridiagonal matrix — a block matrix which is essentially a tridiagonal matrix but with submatrices in place of scalar elements.
• Cartan matrix — a matrix representing a non-semisimple finite-dimensional algebra, or a Lie algebra (note that the two are distinct)
• Cauchy matrix — a matrix whose elements are of the form 1/(x_i + y_j) for (x_i), (y_j) injective sequences.
• Centrosymmetric matrix — a matrix symmetric about its center; i.e., a_ij = a_{n−i+1,n−j+1}
• Circulant matrix — a matrix where each row is a circular shift of its predecessor.
• Commutation matrix — a matrix used for transforming the vectorized form of a matrix into the vectorized form of its transpose.
• Companion matrix — a matrix whose eigenvalues are equal to the roots of the polynomial.
• Conference matrix — a square matrix with zero diagonal and +1 and −1 off the diagonal, such that C^TC is a multiple of the identity matrix.
• Copositive matrix — a square matrix A with real coefficients, such that f(x) = x^TAx is nonnegative for every nonnegative vector x
• Coxeter matrix — a matrix related to Coxeter groups, which describe symmetries in a structure or system.
• Defective matrix — a square matrix that does not have a complete basis of eigenvectors, and is thus not diagonalisable.
• Derogatory matrix — a square n×n matrix whose minimal polynomial is of order less than n.
• Diagonally dominant matrix — a matrix whose entries satisfy
• 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.
• Duplication matrix — a linear transformation matrix used for transforming half-vectorizations of matrices into vectorizations.
• Elementary matrix — a matrix derived by applying an elementary row operation to the identity matrix.
• Elimination matrix — a linear transformation matrix used for transforming vectorizations of matrices into half-vectorizations.
• Equivalent matrix — a matrix that can be derived from another matrix through a sequence of elementary row or column operations.
• Euclidean distance matrix — a matrix that describes the pairwise distances between points in Euclidean space.
• Gell-Mann matrices — a generalisation of the Pauli matrices, these matrices are one notable representation of the infinitesimal generators of the special unitary group, SU(3).
• Generalized permutation matrix — a square matrix with precisely one nonzero element in each row and column.
• Generator matrix — a matrix in coding theory whose rows generate all elements of a linear code.
• Gramian matrix — a real symmetric matrix that can be used to test for linear independence of any function.
• Hadamard matrix — a square matrix with entries +1, −1 whose rows are mutually orthogonal.
• Hankel matrix — a matrix with constant skew-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.
• Hollow matrix — a square matrix whose diagonal comprises only zero elements.
• Householder matrix — a transformation matrix widely used in matrix algorithms.
• Hurwitz matrix — a matrix whose eigenvalues have strictly negative real part. A stable system of differential equations may be represented by a Hurwitz matrix.
• Idempotent matrix — a matrix that has the property A² = A.
• Incidence matrix — a matrix representing a relationship between two classes of objects (used both inside and outside of graph theory).
• Integer matrix — a matrix whose elements are all integers.
• Invertible matrix — a square matrix with a multiplicative inverse.
• Involutary matrix — any square matrix which is its own inverse. Subclasses include the signature matrix and permutation matrix.
• Jacobian matrix — a matrix of first-order partial derivatives of a vector-valued function.
• Logical matrix — a k-dimensional array of boolean values that represents a k-adic relation.
• Moment matrix — a symmetric matrix whose elements are the products of common row/column index dependent monomials.
• Monomial matrix — a square matrix with exactly one non-zero entry in each row and column. Another name for generalized permutation matrix.
• 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 — a matrix whose columns are orthonormal vectors.
• Partitioned matrix — another name for a block matrix (a matrix partitioned into sub-matrices, or equivalently, a matrix whose elements are themselves matrices rather than scalars)
• Payoff matrix — a matrix in game theory, that represents the payoffs in a normal form game where players move simultaneously
• 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 — a 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 — a matrix that occurs in the study of analytical interpolation problems.
• Polynomial matrix — a matrix with polynomials as its elements.
• 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.
• Rotation matrix — a matrix representing a rotational geometric transformation.
• Seifert matrix — a matrix in knot theory, primarily for the algebraic analysis of topological properties of knots and links.
• Shear matrix — an elementary matrix whose corresponding geometric transformation is a shear transformation.
• Signature matrix — a diagonal matrix where the diagonal elements are either +1 or −1.
• 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 — a matrix with relatively few non-zero elements. Sparse matrix algorithms can tackle huge sparse matrices that are utterly impractical for dense matrix algorithms.
• Square matrix — an n by n matrix. The set of all square matrices form an associative algebra with identity.
• Stability matrix — another name for a Hurwitz matrix.
• Stieltjes matrix — an M-matrix which is symmetric and has an inverse.
• 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 coprime 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 — a matrix representing a linear transformation, often from one co-ordinate space to another to facilitate a geometric transform or projection.
• 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.
• X-Y-Z matrix — a generalisation of the (rectangular) matrix to a cuboidal form (a 3-dimensional array of entries).
• Z-matrix — a matrix with all off-diagonal entries less than zero.

[edit] Constant matrices

The list below comprises matrices whose elements are constant for any given dimension (size) of matrix.

• Exchange matrix — a binary matrix with ones on the anti-diagonal, and zeroes everywhere else.
• Hilbert matrix — a Hankel matrix with elements H_ij = (i + j − 1)⁻¹.
• Identity matrix — a square diagonal matrix, with all entries on the main diagonal equal to 1, and the rest 0.
• Lehmer matrix — a positive, symmetric matrix whose elements a_ij are given by min(i,j) ÷ max(i,j).
• Pascal matrix — a matrix containing the entries of Pascal's triangle.
• Pauli matrices — a set of three 2 × 2 complex Hermitian and unitary matrices. When combined with the I₂ identity matrix, they form an orthogonal basis for the 2 × 2 complex Hermitian matrices.
• Shift matrix — a matrix with ones on the superdiagonal or subdiagonal and zeroes elsewhere. Multiplication by it 'shifts' matrix elements by one position.
• Zero matrix — a matrix with all entries equal to zero.

[edit] Matrices used in statistics

The following matrices find their main application in statistics and probability theory.

• Correlation matrix — a symmetric n×n matrix, formed by the pairwise correlation coefficients of several random variables.
• Covariance matrix — a symmetric n×n matrix, formed by the pairwise covariances of several random variables. Sometimes called a dispersion matrix.
• Dispersion matrix — another name for a covariance matrix.
• Doubly stochastic matrix — a non-negative matrix such that each row and each column sums to 1 (thus the matrix is both left stochastic and right stochastic)
• Fisher information matrix — a matrix representing the variance of the partial derivative, with respect to a parameter, of the log of the likelihood function of a random variable.
• Stochastic matrix — a non-negative matrix describing a stochastic process. The sum of entries of any row is one.
• Transition matrix — a matrix representing the probabilities of conditions changing from one state to another in a Markov chain

[edit] Matrices used in graph theory

The following matrices find their main application in graph and network theory.

• Adjacency matrix — a square matrix representing a graph, with a_ij non-zero if vertex i and vertex j are adjacent.
• Biadjacency matrix — a special class of adjacency matrix that describes adjacency in bipartite graphs.
• Degree matrix — a diagonal matrix defining the degree of each vertex in a graph.
• Incidence matrix — a matrix representing a relationship between two classes of objects (usually vertices and edges in the context of graph theory).
• Laplacian matrix — a matrix equal to the degree matrix minus the adjacency matrix for a graph, used to find the number of spanning trees in the graph.
• Seidel adjacency matrix — a matrix similar to the usual adjacency matrix but with −1 for adjacency; +1 for nonadjacency; 0 on the diagonal.

[edit] Matrices used in science and engineering

• Cabibbo-Kobayashi-Maskawa matrix — a unitary matrix used in particle physics to describe the strength of flavour-changing weak decays.
• Density matrix — a matrix describing the statistical state of a quantum system. Hermitian, non-negative and with trace 1.
• Fundamental matrix — a 3 × 3 matrix in computer vision that relates corresponding points in stereo images.
• Fuzzy associative matrix — a matrix in artificial intelligence, used in machine learning processes.
• Hamiltonian matrix — a matrix used in a variety of fields, including quantum mechanics and linear quadratic regulator (LQR) systems.
• Irregular matrix — a matrix used in computer science which has a varying number of elements in each row.
• Overlap matrix — a type of Gramian matrix, used in quantum chemistry to describe the inter-relationship of a set of basis vectors of a quantum system.
• S matrix — a matrix in quantum mechanics that connects asymptotic (infinite past and future) particle states.
• Substitution matrix — a matrix from bioinformatics, which describes mutation rates of amino acid or DNA sequences.
• Z-matrix — a matrix in chemistry, representing a molecule in terms of its relative atomic geometry.

[edit] Other matrix-related terms and definitions

• Jordan canonical form — an 'almost' diagonalised matrix, where the only non-zero elements appear on the lead and super-diagonals.
• Linear independence — two or more vectors are linearly independent if there is no way to construct one from linear combinations of the others.
• Matrix exponential — defined by the exponential series
• Matrix representation of conic sections
• Pseudoinverse — a generalization of the inverse matrix.
• Row echelon form — a matrix in this form is the result of applying the forward elimination procedure to a matrix (as used in Gaussian elimination).
• Wronskian — the determinant of a matrix of functions and their derivatives such that row n is the (n-1)^th derivative of row one.