- List of matrices
This page lists some important classes of matrices used in

mathematics ,science andengineering :**Matrices in mathematics***

(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 ofpermutation matrices that arises fromDodgson 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 — 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.

*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 aLie 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 toCoxeter groups , which describe symmetries in a structure or system.

*Defective matrix — a square matrix that does not have a complete basis ofeigenvectors , and is thus notdiagonalisable .

*Derogatory matrix — a square "n×n" matrix whoseminimal 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 themain diagonal equal to zero.

*Diagonalizable matrix — a square matrix similar to a diagonal matrix. It has a complete set of linearly independenteigenvector s.

*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 inEuclidean 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 thePauli matrices , these matrices are one notable representation of the infinitesimal generators of thespecial unitary group , SU(3).

*Generalized permutation matrix — a square matrix with precisely one nonzero element in each row and column.

*Generator matrix — a matrix incoding theory whose rows generate all elements of alinear code .

*Gramian matrix — a real symmetric matrix that can be used to test forlinear 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 itsconjugate 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 asignature 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 dependentmonomials .

*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 itsconjugate transpose . Normal matrices are precisely the matrices to which thespectral theorem applies.

*Orthogonal matrix — a matrix whose inverse is equal to itstranspose , "A"^{−1}= "A"^{"T"}.

*Orthonormal matrix — a matrix whose columns areorthonormal 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 ingame theory , that represents the payoffs in anormal 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 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., "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 everyeigenvalue 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 inknot 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 itsconjugate transpose , "A"^{*}= −"A".

*Skew-symmetric matrix — a matrix which is equal to the negative of itstranspose , "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 anassociative algebra with identity.

*Stability matrix — another name for aHurwitz matrix .

*Stieltjes matrix — anM-matrix which is symmetric and has an inverse.

*Sylvester matrix — a square matrix whose entries come from coefficients of twopolynomials . The Sylvester matrix is nonsingular if and only if the two polynomials arecoprime to each other.

*Symmetric matrix — a square matrix which is equal to itstranspose , "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 withdeterminant s of all its square submatrices positive. It is used in generating the reference points ofBézier curve incomputer graphics .

*Totally unimodular matrix — a matrix for which every non-singular square submatrix is unimodular. This has some implications in thelinear programming relaxation of aninteger program .

*Transformation matrix — a matrix representing alinear 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 itsconjugate 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 $\{0,1,-1\}$, 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.**Constant matrices**The list below comprises matrices whose elements are constant for any given dimension (size) of matrix.

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Exchange matrix — abinary 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 ofPascal'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 andprobability theory .

*Bernoulli matrix — a square matrix with entries +1, −1, with equalprobability of each.

*Correlation matrix — a symmetric "n×n" matrix, formed by the pairwisecorrelation coefficient s of severalrandom variable s.

*Covariance matrix — a symmetric "n×n" matrix, formed by the pairwisecovariance s 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 — 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 **Matrices used in graph theory**The following matrices find their main application in graph and

network theory .

*Adjacency matrix — a square matrix representing a graph, with "a_{ij}" non-zero if vertex "i" and vertex "j" are adjacent.

*Biadjacency matrix — a special class ofadjacency matrix that describes adjacency inbipartite graph s.

*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 usualadjacency 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 inparticle 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 incomputer vision that relates corresponding points in stereo images.

*Fuzzy associative matrix — a matrix inartificial intelligence , used in machine learning processes.

*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 vector s of a quantum system.

*S matrix — a matrix inquantum mechanics that connects asymptotic (infinite past and future) particle states.

*State Transition matrix — Exponent of state matrix in control systems.

*Substitution matrix — a matrix frombioinformatics , which describes mutation rates ofamino acid orDNA sequences.

*Z-matrix — a matrix inchemistry , 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 fromlinear combination s of the others.

*Matrix exponential — defined by the exponential 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 the "forward elimination" procedure to a matrix (as used inGaussian 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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