This article lists some important classes ofmatrices used inmathematics,science andengineering. Amatrix (plural matrices, or less commonly matrixes) is a rectangulararray ofnumbers calledentries. Matrices have a long history of both study and application, leading to diverse ways of classifying matrices. A first group is matrices satisfying concrete conditions of the entries, including constant matrices. Important examples include theidentity matrix given by

and thezero matrix of dimension${\displaystyle m\times n}$. For example:

.

Further ways of classifying matrices are according to theireigenvalues, or by imposing conditions on theproduct of the matrix with other matrices. Finally, many domains, both in mathematics and other sciences includingphysics andchemistry, have particular matrices that are applied chiefly in these areas.

The list below comprises matrices whose elements are constant for any given dimension (size) of matrix. The matrix entries will be denoteda_ij. The table below uses theKronecker delta δ_ij for two integersi andj which is 1 ifi =j and 0 else.

The following lists matrices whose entries are subject to certain conditions. Many of them apply tosquare matrices only, that is matrices with the same number of columns and rows. Themain diagonal of a square matrix is thediagonal joining the upper left corner and the lower right one or equivalently the entriesa_i,i. The other diagonal is called anti-diagonal (or counter-diagonal).

Name	Explanation	Notes, references
(0,1)-matrix	A matrix with all elements either 0 or 1.	Synonym forbinary matrix orlogical matrix.
Alternant matrix	A matrix in which successive columns have a particular function applied to their entries.
Alternating sign matrix	A square matrix with entries 0, 1 and −1 such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign.
Anti-diagonal matrix	A square matrix with all entries off the anti-diagonal equal to zero.
Anti-Hermitian matrix		Synonym forskew-Hermitian matrix.
Anti-symmetric matrix		Synonym forskew-symmetric matrix.
Arrowhead matrix	A square matrix containing zeros in all entries except for the first row, first column, and main diagonal.
Band matrix	A square matrix whose non-zero entries are confined to a diagonalband.
Bidiagonal matrix	A matrix with elements only on the main diagonal and either the superdiagonal or subdiagonal.	Sometimes defined differently, see article.
Binary matrix	A matrix whose entries are all either 0 or 1.	Synonym for(0,1)-matrix orlogical matrix.[1]
Bisymmetric matrix	A square matrix that is symmetric with respect to its main diagonal and its main cross-diagonal.
Block-diagonal matrix	Ablock 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.
Boolean matrix	A matrix whose entries are taken from aBoolean algebra.
Cauchy matrix	A matrix whose elements are of the form 1/(xi +yj) for (xi), (yj) injective sequences (i.e., taking every value only once).
Centrosymmetric matrix	A matrix symmetric about its center; i.e.,aij = an−i+1,n−j+1.
Circulant matrix	A matrix where each row is a circular shift of its predecessor.
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.
Complex Hadamard matrix	A matrix with all rows and columns mutually orthogonal, whose entries are unimodular.
Compound matrix	A matrix whose entries are generated by the determinants of all minors of a matrix.
Copositive matrix	A square matrixA with real coefficients, such that${\displaystyle f(x)=x^{T}Ax}$ is nonnegative for every nonnegative vectorx
Diagonally dominant matrix	A matrix whose entries satisfy${\displaystyle |a_{ii}|>\sum _{j\ne i}|a_{ij}|}$.
Diagonal matrix	A square matrix with all entries outside themain diagonal equal to zero.
Discrete Fourier-transform matrix	Multiplying by a vector gives the DFT of the vector as result.
Elementary matrix	A square matrix derived by applying an elementary row operation to the identity matrix.
Equivalent matrix	A matrix that can be derived from another matrix through a sequence of elementary row or column operations.
Frobenius matrix	A square matrix in the form of an identity matrix but with arbitrary entries in one column below the main diagonal.
GCD matrix	The${\displaystyle n\times n}$ matrix${\displaystyle (S)}$ having the greatest common divisor${\displaystyle (x_i,x_j)}$ as its${\displaystyle ij}$ entry, where${\displaystyle x_i,x_j\in S}$.
Generalized permutation matrix	A square matrix with precisely one nonzero element in each row and column.
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 itsconjugate transpose,A =A*.
Hessenberg matrix	An "almost" triangular matrix, for example, an upper Hessenberg matrix has zero entries below the first subdiagonal.
Hollow matrix	A square matrix whose main diagonal comprises only zero elements.
Integer matrix	A matrix whose entries are all integers.
Logical matrix	A matrix with all entries either 0 or 1.	Synonym for(0,1)-matrix,binary matrix orBoolean matrix. Can be used to represent ak-adicrelation.
Markov matrix	A matrix of non-negative real numbers, such that the entries in each row sum to 1.
Metzler matrix	A matrix whose off-diagonal entries are non-negative.
Monomial matrix	A square matrix with exactly one non-zero entry in each row and column.	Synonym forgeneralized permutation matrix.
Moore matrix	A row consists ofa,aq,aq², etc., and each row uses a different variable.
Nonnegative matrix	A matrix with all nonnegative entries.
Null-symmetric matrix	A square matrix whose null space (orkernel) is equal to itstranspose, N(A) = N(AT) or ker(A) = ker(AT).	Synonym for kernel-symmetric matrices. Examples include (but not limited to) symmetric, skew-symmetric, and normal matrices.
Null-Hermitian matrix	A square matrix whose null space (orkernel) is equal to itsconjugate transpose, N(A)=N(A*) or ker(A)=ker(A*).	Synonym for kernel-Hermitian matrices. Examples include (but not limited) to Hermitian, skew-Hermitian matrices, and normal matrices.
Partitioned matrix	A matrix partitioned into sub-matrices, or equivalently, a matrix whose entries are themselves matrices rather than scalars.	Synonym forblock matrix.
Parisi matrix	A block-hierarchical matrix. It consist of growing blocks placed along the diagonal, each block is itself a Parisi matrix of a smaller size.	In theory of spin-glasses is also known as a replica matrix.
Pentadiagonal matrix	A matrix with the only nonzero entries on the main diagonal and the two diagonals just above and below the main one.
Permutation matrix	A matrix representation of apermutation, 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.,aij = an−j+1,n−i+1.
Polynomial matrix	A matrix whose entries arepolynomials.
Positive matrix	A matrix with all positive entries.
Quaternionic matrix	A matrix whose entries arequaternions.
Random matrix	A matrix whose entries arerandom variables
Sign matrix	A matrix whose entries are either +1, 0, or −1.
Signature matrix	A diagonal matrix where the diagonal elements are either +1 or −1.
Single-entry matrix	A matrix where a single element is one and the rest of the elements are zero.
Skew-Hermitian matrix	A square matrix which is equal to the negative of itsconjugate transpose,A* = −A.
Skew-symmetric matrix	A matrix which is equal to the negative of itstranspose,AT = −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.
Symmetric matrix	A square matrix which is equal to itstranspose,A =AT (ai,j =aj,i).
Toeplitz matrix	A matrix with constant diagonals.
Totally positive matrix	A matrix withdeterminants of all its square submatrices positive.
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.
X–Y–Z matrix	A generalization to three dimensions of the concept oftwo-dimensional array
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, and the property that the dot product of any two distinct rows (or columns) is zero.
Z-matrix	A matrix with all off-diagonal entries less than zero.

A number of matrix-related notions is about properties of products or inverses of the given matrix. Thematrix product of am-by-n matrixA and an-by-k matrixB is them-by-k matrixC given by

${\displaystyle (C{)}_{i,j}=\sum _{r=1}^n{A}_{i,r}{B}_{r,j}.}$^[2]

This matrix product is denotedAB. Unlike the product of numbers, matrix products are notcommutative, that is to sayAB need not be equal toBA.^[2] A number of notions are concerned with the failure of this commutativity. Aninverse of square matrixA is a matrixB (necessarily of the same dimension asA) such thatAB =I. Equivalently,BA =I. An inverse need not exist. If it exists,B is uniquely determined, and is also calledthe inverse ofA, denotedA⁻¹.

Name	Definition	Comments
Adjugate matrix	Transpose of thecofactor matrix	Theinverse of a matrix is its adjugate matrix divided by itsdeterminant
Augmented matrix	Matrix whose rows are concatenations of the rows of two smaller matrices	Used for performing the samerow operations on two matrices
Bézout matrix	Square matrix whosedeterminant is theresultant of two polynomials	See alsoSylvester matrix
Jabotinsky matrix	Infinite matrix of theTaylor coefficients of ananalytic function and its integer powers	The composition of two functions can be expressed as the product of their Jabotinsky matrices
Cartan matrix	A matrix associated with either a finite-dimensionalassociative algebra, or asemisimple Lie algebra
Cofactor matrix	Formed by thecofactors of a square matrix, that is, the signedminors, of the matrix	Transpose of theAdjugate matrix
Companion matrix	A matrix having the coefficients of a polynomial as last column, and having the polynomial as itscharacteristic polynomial
Coxeter matrix	A matrix which describes the relations between theinvolutions that generate aCoxeter group
Distance matrix	The square matrix formed by the pairwise distances of a set ofpoints	Euclidean distance matrix is a special case
Euclidean distance matrix	A matrix that describes the pairwise distances betweenpoints inEuclidean space	See alsodistance matrix
Fundamental matrix	The matrix formed from the fundamental solutions of asystem of linear differential equations
Generator matrix	InCoding theory, a matrix whose rowsspan alinear code
Gramian matrix	The symmetric matrix of the pairwiseinner products of a set of vectors in aninner product space
Hessian matrix	The square matrix ofsecond partial derivatives of afunction of several variables
Householder matrix	The matrix of areflection with respect to ahyperplane passing through the origin
Jacobian matrix	The matrix of the partial derivatives of afunction of several variables
Moment matrix		Used instatistics andSum-of-squares optimization
Payoff matrix	A matrix ingame theory andeconomics, that represents the payoffs in anormal form game where players move simultaneously
Pick matrix	A matrix that occurs in the study of analytical interpolation problems
Rotation matrix	A matrix representing arotation
Seifert matrix	A matrix inknot theory, primarily for the algebraic analysis of topological properties of knots and links.	Alexander polynomial
Shear matrix	The matrix of ashear transformation
Similarity matrix	A matrix of scores which express the similarity between two data points	Sequence alignment
Sylvester matrix	A square matrix whose entries come from the coefficients of twopolynomials	The Sylvester matrix is nonsingular if and only if the two polynomials arecoprime to each other
Symplectic matrix	The real matrix of asymplectic transformation
Transformation matrix	The matrix of alinear transformation or ageometric transformation
Wedderburn matrix	A matrix of the form${\displaystyle A-(y^{T}Ax{)}^{-1}Ax{y}^{T}A}$, used for rank-reduction & biconjugate decompositions	Analysis of matrix decompositions

The following matrices find their main application instatistics andprobability theory.

• Bernoulli matrix — a square matrix with entries +1, −1, with equalprobability of each.
• Centering matrix — a matrix which, when multiplied with a vector, has the same effect as subtracting the mean of the components of the vector from every component.
• Correlation matrix — a symmetricn×n matrix, formed by the pairwisecorrelation coefficients of severalrandom variables.
• Covariance matrix — a symmetricn×n matrix, formed by the pairwisecovariances of several random variables. Sometimes called adispersion matrix.
• Dispersion matrix — another name for acovariance matrix.
• Doubly stochastic matrix — a non-negative matrix such that each row and each column sums to 1 (thus the matrix is bothleft stochastic andright 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.
• Hat matrix — a square matrix used in statistics to relate fitted values to observed values.
• Orthostochastic matrix — doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some orthogonal matrix
• Precision matrix — a symmetricn×n matrix, formed by inverting thecovariance matrix. Also called theinformation matrix.
• Stochastic matrix — anon-negative matrix describing astochastic process. The sum of entries of any row is one.
• Transition matrix — a matrix representing theprobabilities of conditions changing from one state to another in aMarkov chain
• Unistochastic matrix — a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some unitary matrix

The following matrices find their main application ingraph andnetwork theory.

• Adjacency matrix — a square matrix representing a graph, witha_ij non-zero if vertexi and vertexj are adjacent.
• Biadjacency matrix — a special class ofadjacency matrix that describes adjacency inbipartite graphs.
• Degree matrix — a diagonal matrix defining the degree of eachvertex in a graph.
• Edmonds matrix — a square matrix of a bipartite graph.
• Incidence matrix — a matrix representing a relationship between two classes of objects (usuallyvertices andedges 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 usualadjacency matrix but with −1 for adjacency; +1 for nonadjacency; 0 on the diagonal.
• Skew-adjacency matrix — anadjacency matrix in which each non-zeroa_ij is 1 or −1, accordingly as the directioni → j matches or opposes that of an initially specified orientation.
• Tutte matrix — a generalization of the Edmonds matrix for a balanced bipartite graph.

• Cabibbo–Kobayashi–Maskawa matrix — a unitary matrix used inparticle physics to describe the strength offlavour-changing weak decays.
• Density matrix — a matrix describing the statistical state of a quantum system.Hermitian,non-negative and withtrace 1.
• Fundamental matrix (computer vision) — a 3 × 3 matrix incomputer vision that relates corresponding points in stereo images.
• Fuzzy associative matrix — a matrix inartificial intelligence, used in machine learning processes.
• Gamma matrices — 4 × 4 matrices inquantum field theory.
• Gell-Mann matrices — ageneralization of the Pauli matrices; these matrices are one notable representation of theinfinitesimal generators of thespecial unitary group SU(3).
• Hamiltonian matrix — a matrix used in a variety of fields, includingquantum mechanics andlinear-quadratic regulator (LQR) systems.
• Irregular matrix — a matrix used incomputer science which has a varying number of elements in each row.
• Overlap matrix — a type ofGramian matrix, used inquantum chemistry to describe the inter-relationship of a set ofbasis vectors of aquantum system.
• S matrix — a matrix inquantum mechanics that connects asymptotic (infinite past and future) particle states.
• Scattering matrix - a matrix in Microwave Engineering that describes how the power move in a multiport system.
• State transition matrix — exponent of state matrix in control systems.
• Substitution matrix — a matrix frombioinformatics, which describes mutation rates ofamino acid orDNA sequences.
• Supnick matrix — a square matrix used incomputer science.
• Z-matrix — a matrix inchemistry, representing a molecule in terms of its relative atomic geometry.

• Wilson matrix, a matrix used as an example for test purposes.

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

• Perfect matrix

• Hogben, Leslie (2006),Handbook of Linear Algebra (Discrete Mathematics and Its Applications), Boca Raton: Chapman & Hall/CRC,ISBN 978-1-58488-510-8

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