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

## Constant matrices

## Matrices used in statistics

The following matrices find their main application in statistics and probability theory.## Matrices used in graph theory

The following matrices find their main application in graph and network theory.## Matrices used in science and engineering

## Other matrix-related terms and definitions

- (0,1)-matrix — a matrix with all elements either 0 or 1. Also called a binary matrix.
- Adjugate 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/(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.
- 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 C
^{T}C 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 $f(x)=x^TAx$ 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 |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 A² = 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^{q²}, 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, a², a³, etc., and each row uses a different variable
- Weighing matrix — a square matrix $W$, the entries of which are in $\{0,1,-1\}$, such that $WW^\{T\}=wI$ 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.

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 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

- 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

- 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.

- 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.

- 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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Last updated on Wednesday September 10, 2008 at 09:49:07 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday September 10, 2008 at 09:49:07 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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