# Category: Matrices

Parry–Sullivan invariant
In mathematics, the Parry–Sullivan invariant (or Parry–Sullivan number) is a numerical quantity of interest in the study of incidence matrices in graph theory, and of certain one-dimensional dynamical
Vectorization (mathematics)
In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a column vector. Specifically, the vectorization
Higher spin alternating sign matrix
In mathematics, a higher spin alternating sign matrix is a generalisation of the alternating sign matrix (ASM), where the columns and rows sum to an integer r (the spin) rather than simply summing to
Corner transfer matrix
In statistical mechanics, the corner transfer matrix describes the effect of adding a quadrant to a lattice. Introduced by Rodney Baxter in 1968 as an extension of the Kramers-Wannier row-to-row trans
Involutory matrix
In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, multiplication by the matrix A is an involution if and only if A2 = I, where I is the n × n identity matrix. I
Total active reflection coefficient
The total active reflection coefficient (TARC) within mathematics and physics scattering theory, relates the total incident power to the total outgoing power in an N-port . The TARC is mainly used for
Convergent matrix
In linear algebra, a convergent matrix is a matrix that converges to the zero matrix under matrix exponentiation.
Quincunx matrix
In mathematics, the matrix is sometimes called the quincunx matrix. It is a 2×2 Hadamard matrix, and its rows form the basis of a diagonal square lattice consisting of the integer points whose coordin
Stieltjes matrix
In mathematics, particularly matrix theory, a Stieltjes matrix, named after Thomas Joannes Stieltjes, is a real symmetric positive definite matrix with nonpositive off-diagonal entries. A Stieltjes ma
Q-matrix
In mathematics, a Q-matrix is a square matrix whose associated linear complementarity problem LCP(M,q) has a solution for every vector q.
Square root of a 2 by 2 matrix
A square root of a 2×2 matrix M is another 2×2 matrix R such that M = R2, where R2 stands for the matrix product of R with itself. In general, there can be zero, two, four, or even an infinitude of sq
Hurwitz matrix
In mathematics, a Hurwitz matrix, or Routh–Hurwitz matrix, in engineering stability matrix, is a structured real square matrix constructed with coefficients of a real polynomial.
Aluthge transform
In mathematics and more precisely in functional analysis, the Aluthge transformation is an operation defined on the set of bounded operators of a Hilbert space. It was introduced by to study linear op
CUR matrix approximation
A CUR matrix approximation is a set of three matrices that, when multiplied together, closely approximate a given matrix. A CUR approximation can be used in the same way as the low-rank approximation
Covariance matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covari
Lehmer matrix
In mathematics, particularly matrix theory, the n×n Lehmer matrix (named after Derrick Henry Lehmer) is the constant symmetric matrix defined by Alternatively, this may be written as
Orbital overlap
In chemical bonds, an orbital overlap is the concentration of orbitals on adjacent atoms in the same regions of space. Orbital overlap can lead to bond formation. Linus Pauling explained the importanc
Condition number
In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive
Matrix consimilarity
In linear algebra, two n-by-n matrices A and B are called consimilar if for some invertible matrix , where denotes the elementwise complex conjugation. So for real matrices similar by some real matrix
Companion matrix
In linear algebra, the Frobenius companion matrix of the monic polynomial is the square matrix defined as . Some authors use the transpose of this matrix, which (dually) cycles coordinates, and is mor
Trifocal tensor
In computer vision, the trifocal tensor (also tritensor) is a 3×3×3 array of numbers (i.e., a tensor) that incorporates all projective geometric relationships among three views. It relates the coordin
Nilpotent matrix
In linear algebra, a nilpotent matrix is a square matrix N such that for some positive integer . The smallest such is called the index of , sometimes the degree of . More generally, a nilpotent transf
Wigner D-matrix
The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). It was introduced in 1927 by Eugene Wigner, and plays a fundamental role in the quantum mechanic
Conference matrix
In mathematics, a conference matrix (also called a C-matrix) is a square matrix C with 0 on the diagonal and +1 and −1 off the diagonal, such that CTC is a multiple of the identity matrix I. Thus, if
Packed storage matrix
A packed storage matrix, also known as packed matrix, is a term used in programming for representing an matrix. It is a more compact way than an m-by-n rectangular array by exploiting a special struct
Matrix similarity
In linear algebra, two n-by-n matrices A and B are called similar if there exists an invertible n-by-n matrix P such that Similar matrices represent the same linear map under two (possibly) different
Wilson matrix
Wilson matrix is the following matrix having integers as elements: This is the coefficient matrix of the following system of linear equations considered in a paper by J. Morris published in 1946: Morr
H-matrix (iterative method)
In mathematics, an H-matrix is a matrix whose comparison matrix is an M-matrix. It is useful in iterative methods. Definition: Let A = (aij) be a n × n complex matrix. Then comparison matrix M(A) of c
Mueller calculus
Mueller calculus is a matrix method for manipulating Stokes vectors, which represent the polarization of light. It was developed in 1943 by Hans Mueller. In this technique, the effect of a particular
Cartan matrix
In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first i
Householder transformation
In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane cont
Sylvester matrix
In mathematics, a Sylvester matrix is a matrix associated to two univariate polynomials with coefficients in a field or a commutative ring. The entries of the Sylvester matrix of two polynomials are c
Cross-correlation matrix
The cross-correlation matrix of two random vectors is a matrix containing as elements the cross-correlations of all pairs of elements of the random vectors. The cross-correlation matrix is used in var
Square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order . Any two square matrices of the same order can be added and
Hankel matrix
In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a square matrix in which each ascending skew-diagonal from left to right is constant, e.g.: More generally,
Centering matrix
In mathematics and multivariate statistics, the centering matrix is a symmetric and idempotent matrix, which when multiplied with a vector has the same effect as subtracting the mean of the components
Persymmetric matrix
In mathematics, persymmetric matrix may refer to: 1. * a square matrix which is symmetric with respect to the northeast-to-southwest diagonal; or 2. * a square matrix such that the values on each li
Triangular matrix
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is ca
Pascal matrix
In mathematics, particularly matrix theory and combinatorics, a Pascal matrix is a (possibly infinite) matrix containing the binomial coefficients as its elements. It is thus an encoding of Pascal's t
Brouwer's conjecture
In the mathematical field of spectral graph theory, Brouwer's conjecture is a conjecture by Andries Brouwer on upper bounds for the intermediate sums of the eigenvalues of the Laplacian of a graph in
Skew-Hamiltonian matrix
In linear algebra, skew-Hamiltonian matrices are special matrices which correspond to skew-symmetric bilinear forms on a symplectic vector space. Let V be a vector space, equipped with a symplectic fo
Column groups and row groups
No description available.
Birkhoff polytope
The Birkhoff polytope Bn (also called the assignment polytope, the polytope of doubly stochastic matrices, or the perfect matching polytope of the complete bipartite graph ) is the convex polytope in
Quaternionic matrix
A quaternionic matrix is a matrix whose elements are quaternions.
In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric term
Matrix of ones
In mathematics, a matrix of ones or all-ones matrix is a matrix where every entry is equal to one. Examples of standard notation are given below: Some sources call the all-ones matrix the unit matrix,
Frobenius matrix
A Frobenius matrix is a special kind of square matrix from numerical mathematics. A matrix is a Frobenius matrix if it has the following three properties: * all entries on the main diagonal are ones
Matrix splitting
In the mathematical discipline of numerical linear algebra, a matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. Many iterative methods (for example,
Hessenberg matrix
In linear algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular. To be exact, an upper Hessenberg matrix has zero entries below the first subdiagonal, and a
Modal matrix
In linear algebra, the modal matrix is used in the diagonalization process involving eigenvalues and eigenvectors. Specifically the modal matrix for the matrix is the n × n matrix formed with the eige
Matrix field
In abstract algebra, a matrix field is a field with matrices as elements. In field theory there are two types of fields: finite fields and infinite fields. There are several examples of matrix fields
Cauchy matrix
In mathematics, a Cauchy matrix, named after Augustin-Louis Cauchy, is an m×n matrix with elements aij in the form where and are elements of a field , and and are injective sequences (they contain dis
A complex Hadamard matrix is any complex matrix satisfying two conditions: * unimodularity (the modulus of each entry is unity): * orthogonality: , where denotes the Hermitian transpose of and is th
DFT matrix
In applied mathematics, a DFT matrix is an expression of a discrete Fourier transform (DFT) as a transformation matrix, which can be applied to a signal through matrix multiplication.
Conductance (graph)
In graph theory the conductance of a graph G = (V, E) measures how "well-knit" the graph is: it controls how fast a random walk on G converges to its stationary distribution. The conductance of a grap
Bohemian matrices
A Bohemian matrix family is a set of matrices whosefree entries come from a single discrete, usually finite population, denoted here by P. That is, each entry of any matrix from this particular Bohemi
Jacobian matrix and determinant
In vector calculus, the Jacobian matrix (/dʒəˈkoʊbiən/, /dʒɪ-, jɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is squa
Row and column spaces
In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the imag
Weyl–Brauer matrices
In mathematics, particularly in the theory of spinors, the Weyl–Brauer matrices are an explicit realization of a Clifford algebra as a matrix algebra of 2⌊n/2⌋ × 2⌊n/2⌋ matrices. They generalize the P
Shift matrix
In mathematics, a shift matrix is a binary matrix with ones only on the superdiagonal or subdiagonal, and zeroes elsewhere. A shift matrix U with ones on the superdiagonal is an upper shift matrix. Th
Rosenbrock system matrix
In applied mathematics, the Rosenbrock system matrix or Rosenbrock's system matrix of a linear time-invariant system is a useful representation bridging state-space representation and transfer functio
Diagonalizable matrix
In linear algebra, a square matrix is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix and a diagonal matrix such that , or equi
Augmented matrix
In linear algebra, an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the give
Hermitian matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the
Reducing subspace
In linear algebra, a reducing subspace of a linear map from a Hilbert space to itself is an invariant subspace of whose orthogonal complement is also an invariant subspace of That is, and One says tha
Hilbert matrix
In linear algebra, a Hilbert matrix, introduced by Hilbert, is a square matrix with entries being the unit fractions For example, this is the 5 × 5 Hilbert matrix: The Hilbert matrix can be regarded a
Symplectic matrix
In mathematics, a symplectic matrix is a matrix with real entries that satisfies the condition where denotes the transpose of and is a fixed nonsingular, skew-symmetric matrix. This definition can be
Diagonally dominant matrix
In mathematics, a square matrix is said to be diagonally dominant if, for every row of the matrix, the magnitude of the diagonal entry in a row is larger than or equal to the sum of the magnitudes of
Tutte matrix
In graph theory, the Tutte matrix A of a graph G = (V, E) is a matrix used to determine the existence of a perfect matching: that is, a set of edges which is incident with each vertex exactly once. If
Cross Gramian
In control theory, the cross Gramian is a Gramian matrix used to determine how controllable and observable a linear system is. For the stable time-invariant linear system the cross Gramian is defined
Manin matrix
In mathematics, Manin matrices, named after Yuri Manin who introduced them around 1987–88, are a class of matrices with elements in a not-necessarily commutative ring, which in a certain sense behave
Transformation matrix
In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then for some matrix , called the transformation
Wilkinson matrix
In linear algebra, Wilkinson matrices are symmetric, tridiagonal, order-N matrices with pairs of nearly, but not exactly, equal eigenvalues. It is named after the British mathematician James H. Wilkin
Redheffer star product
In mathematics, the Redheffer star product is a binary operation on linear operators that arises in connection to solving coupled systems of linear equations. It was introduced by Raymond Redheffer in
Vandermonde matrix
In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an m × n matrix or for all indices i and j. Som
S-matrix
In physics, the S-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and
Leslie matrix
The Leslie matrix is a discrete, age-structured model of population growth that is very popular in population ecology named after . The Leslie matrix (also called the Leslie model) is one of the most
Conformable matrix
In mathematics, a matrix is conformable if its dimensions are suitable for defining some operation (e.g. addition, multiplication, etc.).
Circulant matrix
In linear algebra, a circulant matrix is a square matrix in which all row vectors are composed of the same elements and each row vector is rotated one element to the right relative to the preceding ro
R-matrix
The term R-matrix has several meanings, depending on the field of study. The term R-matrix is used in connection with the Yang–Baxter equation. This is an equation which was first introduced in the fi
Transfer function matrix
In control system theory, and various branches of engineering, a transfer function matrix, or just transfer matrix is a generalisation of the transfer functions of single-input single-output (SISO) sy
Shear matrix
In mathematics, a shear matrix or transvection is an elementary matrix that represents the addition of a multiple of one row or column to another. Such a matrix may be derived by taking the identity m
Zero matrix
In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of matrices, and is d
Doubly stochastic matrix
In mathematics, especially in probability and combinatorics, a doubly stochastic matrix (also called bistochastic matrix), is a square matrix of nonnegative real numbers, each of whose rows and column
P-matrix
In mathematics, a P-matrix is a complex square matrix with every principal minor is positive. A closely related class is that of -matrices, which are the closure of the class of P-matrices, with every
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the
Semi-orthogonal matrix
In linear algebra, a semi-orthogonal matrix is a non-square matrix with real entries where: if the number of columns exceeds the number of rows, then the rows are orthonormal vectors; but if the numbe
UK Molecular R-matrix Codes
The UK Molecular R-Matrix codes are a set of software routines used to calculate the effects of collision of electrons with atoms and molecules. The R-matrix method is used in computational quantum me
Plücker matrix
The Plücker matrix is a special skew-symmetric 4 × 4 matrix, which characterizes a straight line in projective space. The matrix is defined by 6 Plücker coordinates with 4 degrees of freedom. It is na
Jacket matrix
In mathematics, a jacket matrix is a of order n if its entries are non-zero and real, complex, or from a finite field, and where In is the identity matrix, and where T denotes the transpose of the mat
Z-matrix (mathematics)
In mathematics, the class of Z-matrices are those matrices whose off-diagonal entries are less than or equal to zero; that is, the matrices of the form: Note that this definition coincides precisely w
Continuant (mathematics)
In algebra, the continuant is a multivariate polynomial representing the determinant of a tridiagonal matrix and having applications in generalized continued fractions.
Anti-diagonal matrix
In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗), known as the anti
Distance matrix
In mathematics, computer science and especially graph theory, a distance matrix is a square matrix (two-dimensional array) containing the distances, taken pairwise, between the elements of a set. Depe
Matrix regularization
In the field of statistical learning theory, matrix regularization generalizes notions of vector regularization to cases where the object to be learned is a matrix. The purpose of regularization is to
Boolean matrix
In mathematics, a Boolean matrix is a matrix with entries from a Boolean algebra. When the two-element Boolean algebra is used, the Boolean matrix is called a logical matrix. (In some contexts, partic
Sylvester equation
In mathematics, in the field of control theory, a Sylvester equation is a matrix equation of the form: Then given matrices A, B, and C, the problem is to find the possible matrices X that obey this eq
Drazin inverse
In mathematics, the Drazin inverse, named after Michael P. Drazin, is a kind of generalized inverse of a matrix. Let A be a square matrix. The index of A is the least nonnegative integer k such that r
Weakly chained diagonally dominant matrix
In mathematics, the weakly chained diagonally dominant matrices are a family of nonsingular matrices that include the strictly diagonally dominant matrices.
Redheffer matrix
In mathematics, a Redheffer matrix, often denoted as studied by , is a square (0,1) matrix whose entries aij are 1 if i divides j or if j = 1; otherwise, aij = 0. It is useful in some contexts to expr
Krawtchouk matrices
In mathematics, Krawtchouk matrices are matrices whose entries are values of Krawtchouk polynomials at nonnegative integer points. The Krawtchouk matrix K(N) is an (N + 1) × (N + 1) matrix. The first
Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a prop
Permutation matrix
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say P, rep
Autocorrelation matrix
No description available.
Magic square
In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. T
Density matrix embedding theory
The density matrix embedding theory (DMET) is a numerical technique to solve strongly correlated electronic structure problems. By mapping the system to a fragment plus its entangled quantum bath, the
Decomposition matrix
In mathematics, and in particular modular representation theory, a decomposition matrix is a matrix that results from writing the irreducible ordinary characters in terms of the irreducible modular ch
Matrix difference equation
A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related to its own value at one or more previous poin
Irregular matrix
An irregular matrix, or ragged matrix, is a matrix that has a different number of elements in each row. Ragged matrices are not used in linear algebra, since standard matrix transformations cannot be
Unimodular polynomial matrix
In mathematics, a unimodular polynomial matrix is a square polynomial matrix whose inverse exists and is itself a polynomial matrix. Equivalently, a polynomial matrix A is unimodular if its determinan
In mathematics a regular Hadamard matrix is a Hadamard matrix whose row and column sums are all equal. While the order of a Hadamard matrix must be 1, 2, or a multiple of 4, regular Hadamard matrices
Scatter matrix
In multivariate statistics and probability theory, the scatter matrix is a statistic that is used to make estimates of the covariance matrix, for instance of the multivariate normal distribution.
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (σ), they
Orthostochastic matrix
In mathematics, an orthostochastic matrix is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some orthogonal matrix. The detailed definition is as fol
Levinson recursion
Levinson recursion or Levinson–Durbin recursion is a procedure in linear algebra to recursively calculate the solution to an equation involving a Toeplitz matrix. The algorithm runs in Θ(n2) time, whi
Block matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block ma
Unistochastic matrix
In mathematics, a unistochastic matrix (also called unitary-stochastic) is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some unitary matrix. A squa
Generalized permutation matrix
In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each co
Kernel (linear algebra)
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map L
Bicomplex number
In abstract algebra, a bicomplex number is a pair (w, z) of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate , and the product of two bicomplex numbers as
EP matrix
In mathematics, an EP matrix (or range-Hermitian matrix or RPN matrix) is a square matrix A whose range is equal to the range of its conjugate transpose A*. Another equivalent characterization of EP m
Constrained generalized inverse
In linear algebra, a constrained generalized inverse is obtained by solving a system of linear equations with an additional constraint that the solution is in a given subspace. One also says that the
Matrix representation
Matrix representation is a method used by a computer language to store matrices of more than one dimension in memory.Fortran and C use different schemes for their native arrays. Fortran uses "Column M
Productive matrix
In linear algebra, a square nonnegative matrix of order is said to be productive, or to be a Leontief matrix, if there exists a nonnegative column matrix such as is a positive matrix.
Generalized inverse
In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element x is an element y that has some properties of an inverse element but not necessarily all of them. Genera
Identity matrix
In linear algebra, the identity matrix of size is the square matrix with ones on the main diagonal and zeros elsewhere.
Supnick matrix
A Supnick matrix or Supnick array – named after Fred Supnick of the City College of New York, who introduced the notion in 1957 – is a Monge array which is also a symmetric matrix.
Exchange matrix
In mathematics, especially linear algebra, the exchange matrices (also called the reversal matrix, backward identity, or standard involutory permutation) are special cases of permutation matrices, whe
In mathematics, in graph theory, the Seidel adjacency matrix of a simple undirected graph G is a symmetric matrix with a row and column for each vertex, having 0 on the diagonal, −1 for positions whos
Variation diminishing property
In mathematics, the variation diminishing property of certain mathematical objects involves diminishing the number of changes in sign (positive to negative or vice versa).
Centrosymmetric matrix
In mathematics, especially in linear algebra and matrix theory, a centrosymmetric matrix is a matrix which is symmetric about its center. More precisely, an n×n matrix A = [Ai,j] is centrosymmetric wh
Fundamental matrix (linear differential equation)
In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations is a matrix-valued function whose columns are linearly independent solutions of the system.Then
Bisymmetric matrix
In mathematics, a bisymmetric matrix is a square matrix that is symmetric about both of its main diagonals. More precisely, an n × n matrix A is bisymmetric if it satisfies both A = AT and AJ = JA whe
Copositive matrix
In mathematics, specifically linear algebra, a real matrix A is copositive if for every nonnegative vector . The collection of all copositive matrices is a proper cone; it includes as a subset the col
Birkhoff algorithm
Birkhoff's algorithm (also called Birkhoff-von-Neumann algorithm) is an algorithm for decomposing a bistochastic matrix into a convex combination of permutation matrices. It was published by Garrett B
Moore determinant of a Hermitian matrix
In mathematics, the Moore determinant is a determinant defined for Hermitian matrices over a quaternion algebra, introduced by Moore.
Nekrasov matrix
In mathematics, a Nekrasov matrix or generalised Nekrasov matrix is a type of diagonally dominant matrix (i.e. one in which the diagonal elements are in some way greater than some function of the non-
Conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugate on each entry (the complex conj
Perfect matrix
In mathematics, a perfect matrix is an m-by-n binary matrix that has no possible k-by-k submatrix K that satisfies the following conditions: * k > 3 * the row and column sums of K are each equal to
Fuzzy associative matrix
A fuzzy associative matrix expresses fuzzy logic rules in tabular form. These rules usually take two variables as input, mapping cleanly to a two-dimensional matrix, although theoretically a matrix of
Toeplitz matrix
In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant. For instance, the following m
Bézout matrix
In mathematics, a Bézout matrix (or Bézoutian or Bezoutiant) is a special square matrix associated with two polynomials, introduced by James Joseph Sylvester and Arthur Cayley and named after Étienne
Idempotent matrix
In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix is idempotent if and only if . For this product to be defined, must necessarily
Moment matrix
In mathematics, a moment matrix is a special symmetric square matrix whose rows and columns are indexed by monomials. The entries of the matrix depend on the product of the indexing monomials only (cf
L-matrix
In mathematics, the class of L-matrices are those matrices whose off-diagonal entries are less than or equal to zero and whose diagonal entries are positive; that is, an L-matrix L satisfies
Hierarchical matrix
In numerical mathematics, hierarchical matrices (H-matrices)are used as data-sparse approximations of non-sparse matrices. While a sparse matrix of dimension can be represented efficiently in units of
Integer matrix
In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix, and the adjacency matrices use
In mathematics, a complex Hadamard matrix H of size N with all its columns (rows) mutually orthogonal, belongs to the Butson-type H(q, N) if all its elements are powers of q-th root of unity,
Green's matrix
In mathematics, and in particular ordinary differential equations, a Green's matrix helps to determine a particular solution to a first-order inhomogeneous linear system of ODEs. The concept is named
Hessian matrix
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of man
Design matrix
In statistics and in particular in regression analysis, a design matrix, also known as model matrix or regressor matrix and often denoted by X, is a matrix of values of explanatory variables of a set
Matrix analysis
In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular topics out of many include; operations defined
Metzler matrix
In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero): It is named after the American economist Lloyd Metzler. Metzler m
Unitary matrix
In linear algebra, a complex square matrix U is unitary if its conjugate transpose U* is also its inverse, that is, if where I is the identity matrix. In physics, especially in quantum mechanics, the
Balanced matrix
In mathematics, a balanced matrix is a 0-1 matrix (a matrix where every entry is either zero or one) that does not contain any square submatrix of odd order having all row sums and all column sums equ
In the Standard Model of particle physics, the Cabibbo–Kobayashi–Maskawa matrix, CKM matrix, quark mixing matrix, or KM matrix is a unitary matrix which contains information on the strength of the fla
Logical matrix
A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1) matrix is a matrix with entries from the Boolean domain B = {0, 1}. Such a matrix can be used to represent a binary relation
Definite matrix
In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector where is the transpose of . More generally, a Hermitian ma
John Williamson (mathematician)
John Williamson (23 May 1901 – 1949) was a Scottish mathematician who worked in the fields of algebra, invariant theory, and linear algebra. Among other contributions, he is known for the Williamson c
Cross-covariance matrix
In probability theory and statistics, a cross-covariance matrix is a matrix whose element in the i, j position is the covariance between the i-th element of a random vector and j-th element of another
Transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix, often d
BLOSUM
In bioinformatics, the BLOSUM (BLOcks SUbstitution Matrix) matrix is a substitution matrix used for sequence alignment of proteins. BLOSUM matrices are used to score alignments between evolutionarily
Birkhoff factorization
In mathematics, Birkhoff factorization or Birkhoff decomposition, introduced by George David Birkhoff, is the factorization of an invertible matrix M with coefficients that are Laurent polynomials in
Gram matrix
In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product .
Next-generation matrix
In epidemiology, the next-generation matrix is used to derive the basic reproduction number, for a compartmental model of the spread of infectious diseases. In population dynamics it is used to comput
Pseudo-determinant
In linear algebra and statistics, the pseudo-determinant is the product of all non-zero eigenvalues of a square matrix. It coincides with the regular determinant when the matrix is non-singular.
Higher-dimensional gamma matrices
In mathematical physics, higher-dimensional gamma matrices generalize to arbitrary dimension the four-dimensional Gamma matrices of Dirac, which are a mainstay of relativistic quantum mechanics. They
Nonnegative matrix
In mathematics, a nonnegative matrix, written is a matrix in which all the elements are equal to or greater than zero, that is, A positive matrix is a matrix in which all the elements are strictly gre
Projection matrix
In statistics, the projection matrix , sometimes also called the influence matrix or hat matrix , maps the vector of response values (dependent variable values) to the vector of fitted values (or pred
DE-9IM
The Dimensionally Extended 9-Intersection Model (DE-9IM) is a topological model and a standard used to describe the spatial relations of two regions (two geometries in two-dimensions, R2), in geometry
Rotation matrix
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix rotates points in the xy plan
Commutation matrix
In mathematics, especially in linear algebra and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix into the vectorized form of its transpose. Specifically,
Brahmagupta matrix
In mathematics, the following matrix was given by Indian mathematician Brahmagupta: It satisfies Powers of the matrix are defined by The and are called . The Brahmagupta matrices can be extended to ne
Compound matrix
In linear algebra, a branch of mathematics, a (multiplicative) compound matrix is a matrix whose entries are all minors, of a given size, of another matrix. Compound matrices are closely related to ex
Polyconvex function
In mathematics, the notion of polyconvexity is a generalization of the notion of convexity for functions defined on spaces of matrices. Let Mm×n(K) denote the space of all m × n matrices over the fiel
Gell-Mann matrices
The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3×3 traceless Hermitian matrices used in the study of the strong interaction in particle physics.They spa
Bartels–Stewart algorithm
In numerical linear algebra, the Bartels–Stewart algorithm is used to numerically solve the Sylvester matrix equation . Developed by R.H. Bartels and G.W. Stewart in 1971, it was the first numerically
WAIFW matrix
In infectious disease modelling, a who acquires infection from whom (WAIFW) matrix is a matrix that describes the rate of transmission of infection between different groups in a population, such as pe
M-matrix
In mathematics, especially linear algebra, an M-matrix is a Z-matrix with eigenvalues whose real parts are nonnegative. The set of non-singular M-matrices are a subset of the class of P-matrices, and
Duplication and elimination matrices
In mathematics, especially in linear algebra and matrix theory, the duplication matrix and the elimination matrix are linear transformations used for transforming half-vectorizations of matrices into
Main diagonal
In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix is the list of entries where . All off-diagonal el
Signature matrix
In mathematics, a signature matrix is a diagonal matrix whose diagonal elements are plus or minus 1, that is, any matrix of the form: Any such matrix is its own inverse, hence is an involutory matrix.
Hadamard's maximal determinant problem, named after Jacques Hadamard, asks for the largest determinant of a matrix with elements equal to 1 or −1. The analogous question for matrices with elements equ
Skew-Hermitian matrix
In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix is skew-He
Laplacian matrix
In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Name
List of named matrices
This article lists some important classes of matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entri
Paley construction
In mathematics, the Paley construction is a method for constructing Hadamard matrices using finite fields. The construction was described in 1933 by the English mathematician Raymond Paley. The Paley
Alternating sign matrix
In mathematics, an alternating sign matrix is a square matrix of 0s, 1s, and −1s such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. These m
Sample mean and covariance
The sample mean (or "empirical mean") and the sample covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or mean value) of a
Substitution matrix
In bioinformatics and evolutionary biology, a substitution matrix describes the frequency at which a character in a nucleotide sequence or a protein sequence changes to other character states over evo
Unimodular matrix
In mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix N t
Walsh matrix
In mathematics, a Walsh matrix is a specific square matrix of dimensions 2n, where n is some particular natural number. The entries of the matrix are either +1 or −1 and its rows as well as columns ar
Incidence matrix
In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is X and the second is Y, t
Linear group
In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication. A linear group is a group that is isomorphic to a m
Comparison matrix
In linear algebra, let A = (aij) be a n × n complex matrix. The comparison matrix M(A) = (αij) of complex matrix A is defined as
Jones calculus
In optics, polarized light can be described using the Jones calculus, discovered by R. C. Jones in 1941. Polarized light is represented by a Jones vector, and linear optical elements are represented b
Hasse–Witt matrix
In mathematics, the Hasse–Witt matrix H of a non-singular algebraic curve C over a finite field F is the matrix of the Frobenius mapping (p-th power mapping where F has q elements, q a power of the pr
Row and column vectors
In linear algebra, a column vector is a column of entries, for example, which may also be viewed as an matrix for some .Similarly, a row vector is a row of entries,or equivalently a matrix for some .
Gamma matrices
In mathematical physics, the gamma matrices, , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representat
Hamiltonian matrix
In mathematics, a Hamiltonian matrix is a 2n-by-2n matrix A such that JA is symmetric, where J is the skew-symmetric matrix and In is the n-by-n identity matrix. In other words, A is Hamiltonian if an
Matrix chain multiplication
Matrix chain multiplication (or the matrix chain ordering problem) is an optimization problem concerning the most efficient way to multiply a given sequence of matrices. The problem is not actually to
Woodbury matrix identity
In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max A. Woodbury, says that the inverse of a rank-k correction of some matrix can be computed by doing a rank-k c
Moore matrix
In linear algebra, a Moore matrix, introduced by E. H. Moore, is a matrix defined over a finite field. When it is a square matrix its determinant is called a Moore determinant (this is unrelated to th
Polynomial matrix
In mathematics, a polynomial matrix or matrix of polynomials is a matrix whose elements are univariate or multivariate polynomials. Equivalently, a polynomial matrix is a polynomial whose coefficients
Brandt matrix
In mathematics, Brandt matrices are matrices, introduced by Brandt, that are related to the number of ideals of given norm in an ideal class of a definite quaternion algebra over the rationals, and th
Matrix congruence
In mathematics, two square matrices A and B over a field are called congruent if there exists an invertible matrix P over the same field such that PTAP = B where "T" denotes the matrix transpose. Matr
Block reflector
"A block reflector is an orthogonal, symmetric matrix that reverses a subspace whose dimension may be greater than one." It is built out of many elementary reflectors. It is also referred to as a tria
Degree matrix
In the mathematical field of algebraic graph theory, the degree matrix of an undirected graph is a diagonal matrix which contains information about the degree of each vertex—that is, the number of edg
Row equivalence
In linear algebra, two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. Alternatively, two m × n matrices are row equivalent if and only if th
Monotone matrix
A real square matrix is monotone (in the sense of Collatz) if for all real vectors , implies , where is the element-wise order on .
Symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a
Stochastic matrix
In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also calle
In the mathematical field of linear algebra, an arrowhead matrix is a square matrix containing zeros in all entries except for the first row, first column, and main diagonal, these entries can be any
Orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is where QT is the transpose of Q and I
Givens rotation
In numerical linear algebra, a Givens rotation is a rotation in the plane spanned by two coordinates axes. Givens rotations are named after Wallace Givens, who introduced them to numerical analysts in
Fock matrix
In the Hartree–Fock method of quantum mechanics, the Fock matrix is a matrix approximating the single-electron energy operator of a given quantum system in a given set of basis vectors.It is most ofte
Transpositions matrix
Transpositions matrix (Tr matrix) is square matrix, , , which elements are obtained from the elements of given n-dimensional vector as follows: , where denotes operation "bitwise Exclusive or" (XOR).
Matrix equivalence
In linear algebra, two rectangular m-by-n matrices A and B are called equivalent if for some invertible n-by-n matrix P and some invertible m-by-m matrix Q. Equivalent matrices represent the same line
Alternant matrix
In linear algebra, an alternant matrix is a matrix formed by applying a finite list of functions pointwise to a fixed column of inputs. An alternant determinant is the determinant of a square alternan
Hollow matrix
In mathematics, a hollow matrix may refer to one of several related classes of matrix: a sparse matrix; a matrix with a large block of zeroes; or a matrix with diagonal entries all zero.
Skew-symmetric matrix
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In
Completely-S matrix
In linear algebra, a completely-S matrix is a square matrix such that for every principal submatrix R there exists a positive vector u such that Ru > 0.
Supermatrix
In mathematics and theoretical physics, a supermatrix is a Z2-graded analog of an ordinary matrix. Specifically, a supermatrix is a 2×2 block matrix with entries in a superalgebra (or superring). The
Hessian automatic differentiation
In applied mathematics, Hessian automatic differentiation are techniques based on automatic differentiation (AD)that calculate the second derivative of an -dimensional function, known as the Hessian m
Invertible matrix
In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate), if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix a
Algebraic Riccati equation
An algebraic Riccati equation is a type of nonlinear equation that arises in the context of infinite-horizon optimal control problems in continuous time or discrete time. A typical algebraic Riccati e
Euclidean distance matrix
In mathematics, a Euclidean distance matrix is an n×n matrix representing the spacing of a set of n points in Euclidean space.For points in k-dimensional space ℝk, the elements of their Euclidean dist