# List of matrices

List of matrices

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

Matrices in mathematics

*(0,1)-matrix — a matrix with all elements either 0 or 1. Also called a "binary matrix".
* Alternant matrix — a matrix in which successive columns have a particular function applied to their entries.
*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".
*Arrowhead matrix — a square matrix containing zeros in all entries except for the first row, first column, and main diagonal
*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).
*Bisymmetric matrix — a square matrix that is symmetric with respect to its main diagonal and its main cross-diagonal.
*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.
*Carleman matrix — a matrix that converts composition of functions to multiplication of matrices
*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/("xi" + "yj") for ("xi"), ("yj") 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.
*Cofactor matrix
*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.
*Complex Hadamard matrix - a maxtrix with all rows and columns mutually orthogonal, whose entries are unimodular.
*Conference matrix — a square matrix with zero diagonal and +1 and −1 off the diagonal, such that CTC is a multiple of the identity matrix.
*Congruent matrix - two matrices "A" and "B" are said congruent if there exists an invertible matrix "P" such that "P"T "A" "P" = "B"
*Copositive matrix — a square matrix "A" with real coefficients, such that $f\left(x\right)=x^TAx$ 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 |"a""ii"| > Σ"j"≠"i" |"a""ij"|.
*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.
*Frobenius matrix — a matrix in the form of an identity matrix but with arbitrary entries in one column below the main diagonal
*Fundamental matrix (linear differential equation)
*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.
*Hat matrix - a square matrix used in statistics to relate fitted values to observed values.
*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 "A"² = "AA" = "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, such as a signature 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".
*Moore matrix — a row consists of 1, "a", "a""q", "a""q"², etc., and each row uses a different variable
*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, a square matrix with exactly one 1 in each row and column, and all other elements 0.
*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.
*Sign matrix — a matrix whose elements are either +1, 0, or −1.
*Signature matrix — a diagonal matrix where the diagonal elements are either +1 or −1.
*Similar matrix — two matrices "A" and "B" are called similar if there exists an invertible matrix "P" such that "P"−1"AP" = "B".
*Similarity matrix — a matrix of scores which express the similarity between two data points.
*Singular matrix — a noninvertible square matrix.
*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".
*Skyline matrix — a rearrangement of the entries of a banded matrix which requires less space.
*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 ("a""i","j" = "a""j","i" ).
*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.
*Unipotent matrix — a square matrix with all eigenvalues equal to 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", "a"², "a"³, etc., and each row uses a different variable
*Weighing matrix — a square matrix $W$, the entries of which are in $\left\{0,1,-1\right\}$, such that $WW^\left\{T\right\}=wI$ for some positive integer $w$.
*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.

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 "aij" 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 "I"2 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.
*Matrix of ones — a matrix with all entries equal to one

Matrices used in statistics

The following matrices find their main application in statistics and probability theory.
*Bernoulli matrix — a square matrix with entries +1, −1, with equal probability of each.
*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.
*Precision matrix — a symmetric "n×n" matrix, formed by inverting the "covariance matrix". Also called the "information matrix".
*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

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 "aij" non-zero if vertex "i" and vertex "j" are adjacent.
*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.

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 (computer vision) — 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.
*State Transition matrix — Exponent of state matrix in control systems.
*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.

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.

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