Editing List of named matrices

{{Short description|None}}
[[Image:Taxonomy of Complex Matrices.svg|thumb|247px|right|Several important classes of matrices are subsets of each other.]]
This article lists some important classes of [[matrix (mathematics)|matrices]] used in [[mathematics]], [[science]] and [[engineering]]. A '''matrix''' (plural matrices, or less commonly matrixes) is a rectangular [[Array data structure|array]] of [[number]]s called ''entries''. 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 the [[identity matrix]] given by
:<math>
I_n = \begin{bmatrix}
1 & 0 & \cdots & 0 \\
0 & 1 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & 1 \end{bmatrix}.</math>
and the [[zero matrix]] of dimension <math>m \times n</math>. For example:

:<math>
O_{2 \times 3} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0  \end{pmatrix}</math>.

Further ways of classifying matrices are according to their [[eigenvalue]]s, or by imposing conditions on the [[matrix product|product]] of the matrix with other matrices. Finally, many domains, both in mathematics and other sciences including [[physics]] and [[chemistry]], have particular matrices that are applied chiefly in these areas.
<!-- !!!!!! PLEASE NOTE! The — dash used is not the usual hyphen. It is an 'em dash' — please either copy and paste an existing one, or use the Insert facility just below the "Save page" button. -->

== Constant matrices ==

The list below comprises matrices whose elements are constant for any given dimension (size) of matrix. The matrix entries will be denoted ''a<sub>ij</sub>''. The table below uses the [[Kronecker delta]] δ<sub>''ij''</sub> for two integers ''i'' and ''j'' which is 1 if ''i'' = ''j'' and 0 else.

{| class="wikitable sortable"
! Name !! Explanation !! Symbolic description of the entries !! Notes
|-
|  [[Commutation matrix]]  || The matrix of the [[linear map]] that maps a matrix to its transpose ||  || See [[vectorization (mathematics)|Vectorization]]
|-
|  [[Duplication matrix]] || The matrix of the linear map mapping the vector of the distinct entries of a [[symmetric matrix]] to the vector of all entries of the matrix|| || See [[vectorization (mathematics)|Vectorization]]
|-
|  [[Elimination matrix]] || The matrix of the linear map mapping the vector of the entries of a matrix to the vector of a part of the entries (for example the vector of the entries that are not below the main diagonal)|| || See [[Vectorization (mathematics)|vectorization]]
|-
| [[Exchange&nbsp;matrix]] || The [[binary matrix]] with ones on the anti-diagonal, and zeroes everywhere else. || ''a<sub>ij</sub>'' = δ<sub>''n''+1−''i'',''j''</sub> || A [[permutation matrix]].
|-
| [[Hilbert matrix]] || || ''a''<sub>''ij''</sub>&nbsp;=&nbsp;(''i''&nbsp;+&nbsp;''j''&nbsp;−&nbsp;1)<sup>−1</sup>. || A [[Hankel matrix]].
|-
| [[Identity matrix]] || A square diagonal matrix, with all entries on the main diagonal equal to 1, and the rest 0. || ''a<sub>ij</sub>'' = δ<sub>''ij''</sub> ||
|-
| [[Lehmer matrix]] || || ''a<sub>ij</sub>'' = min(''i'', ''j'') ÷ max(''i'', ''j''). || A [[positive matrix|positive]] [[symmetric matrix]].
|-
| [[Matrix of ones]] || A matrix with all entries equal to one. || ''a<sub>ij</sub>'' = 1. ||
|-
| [[Pascal matrix]] || A  matrix containing the entries of [[Pascal's triangle]]. ||  ||
|-
| [[Pauli matrices]] || A set of three 2&nbsp;×&nbsp;2 complex Hermitian and unitary matrices. When combined with the ''I''<sub>2</sub> identity matrix, they form an orthogonal basis for the 2&nbsp;×&nbsp;2 complex Hermitian matrices. || ||
|-
| [[Redheffer&nbsp;matrix]] || Encodes a [[Dirichlet convolution]]. Matrix entries are given by the [[divisor function]]; entires of the inverse are given by the [[Möbius function]]. || ''a''<sub>''ij''</sub> are 1 if ''i'' divides ''j'' or if ''j'' = 1; otherwise, ''a''<sub>''ij''</sub> = 0. || A (0, 1)-matrix.
|-
| [[Shift matrix]] || A matrix with ones on the superdiagonal or subdiagonal and zeroes elsewhere. || ''a<sub>ij</sub>'' = δ<sub>''i''+1,''j''</sub> or ''a<sub>ij</sub>'' = δ<sub>''i''−1,''j''</sub> || Multiplication by it shifts matrix elements by one position.
|-
| [[Zero matrix]] || A matrix with all entries equal to zero. || ''a<sub>ij</sub>'' = 0. ||
|}

== Specific patterns for entries ==
The following lists matrices whose entries are subject to certain conditions. Many of them apply to ''square matrices'' only, that is matrices with the same number of columns and rows. The [[main diagonal]] of a square matrix is the [[diagonal]] joining the upper left corner and the lower right one or equivalently the entries ''a''<sub>''i'',''i''</sub>. The other diagonal is called anti-diagonal (or counter-diagonal).

{| class="wikitable sortable"
! Name !! Explanation !! Notes, references
|-
| [[Logical matrix|(0,1)-matrix]] || A matrix with all elements either 0 or 1. || Synonym for ''binary matrix'' or ''logical 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 &minus;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. ||
|-
| [[Skew-Hermitian matrix|Anti-Hermitian matrix]] ||  || Synonym for ''skew-Hermitian matrix''.
|-
| [[Skew-symmetric matrix|Anti-symmetric matrix]] ||  || Synonym for ''skew-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 diagonal ''band''. ||
|-
| [[Bidiagonal matrix]] || A matrix with elements only on the main diagonal and either the superdiagonal or subdiagonal. || Sometimes defined differently, see article.
|-
| [[Logical matrix|Binary matrix]] || A matrix whose entries are all either 0 or 1. || Synonym for ''(0,1)-matrix'' or ''logical matrix''.<ref>{{Harvard citations |last1=Hogben |nb=yes |loc=Ch. 31.3 |year=2006}}.</ref>
|-
| [[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. ||
|-
| [[Boolean matrix]] || A matrix whose entries are taken from a [[Boolean algebra]]. ||
|-
| [[Cauchy matrix]] || A matrix whose elements are of the form 1/(''x<sub>i</sub>'' + ''y<sub>j</sub>'') for (''x<sub>i</sub>''), (''y<sub>j</sub>'') injective sequences (i.e., taking every value only once). ||
|-
| [[Centrosymmetric matrix]] || A matrix symmetric about its center; i.e., ''a''<sub>''ij''</sub>&nbsp;=&nbsp;''a''<sub>''n''−''i''+1,''n''−''j''+1</sub>. ||
|-
|  [[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 C<sup>T</sup>C 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 matrix ''A'' with real coefficients, such that <math>f(x) = x^T A x</math> is nonnegative for every nonnegative vector ''x'' ||
|-
| [[Diagonally dominant matrix]] || A matrix whose entries satisfy <math>|a_{ii}| > \sum_{j \ne i} |a_{ij}|</math>. ||
|-
| [[Diagonal matrix]] || A square matrix with all entries outside the [[main diagonal]] equal to zero. ||
|-
| [[DFT matrix|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 <math>n\times n</math> matrix <math>(S)</math> having the greatest common divisor <math>(x_i, x_j)</math> as its <math>ij</math> entry, where <math>x_i, x_j\in S</math>. ||
|-
| [[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 its [[conjugate transpose]], ''A'' = ''A''<sup>*</sup>.||
|-
| [[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'' or ''Boolean matrix''. Can be used to represent a ''k''-adic [[relation (mathematics)|relation]].
|-
| [[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. ||
|-
| [[Generalized permutation matrix|Monomial matrix]] || A square matrix with exactly one non-zero entry in each row and column. || Synonym for ''generalized permutation matrix''.
|-
| [[Moore matrix]] ||A row consists of ''a'', ''a''<sup>''q''</sup>, ''a''<sup>''q''²</sup>, 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 (or [[Kernel (linear algebra)|kernel]]) is equal to its [[transpose]], N(''A)'' = N(''A<sup>T</sup>'') or ker(''A)'' = ker(''A<sup>T</sup>'').
|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 (or [[Kernel (linear algebra)|kernel]]) is equal to its [[conjugate transpose]], N(''A'')=N(''A''<sup>*</sup>) or ker(''A'')=ker(''A''<sup>*</sup>).
|Synonym for kernel-Hermitian matrices.  Examples include (but not limited) to Hermitian, skew-Hermitian matrices, and normal matrices.
|-
| [[Block matrix|Partitioned matrix]] || A matrix partitioned into sub-matrices, or equivalently, a matrix whose entries are themselves matrices rather than scalars. || Synonym for ''block 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 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''<sub>''ij''</sub>&nbsp;=&nbsp;''a''<sub>''n''−''j''+1,''n''−''i''+1</sub>. ||
|-
| [[Polynomial matrix]] || A matrix whose entries are [[polynomial]]s. ||
|-
| [[Positive matrix]] || A matrix with all positive entries. ||
|-
| [[Quaternionic matrix]] || A matrix whose entries are [[quaternion]]s. ||
|-
|  [[Random matrix]] || A matrix whose entries are [[random variable]]s ||
|-
| [[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 its [[conjugate transpose]], ''A''<sup>*</sup> = −''A''. ||
|-
| [[Skew-symmetric matrix]] || A matrix which is equal to the negative of its [[transpose]], ''A''<sup>T</sup> = −''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 its [[transpose]], ''A'' = ''A''<sup>T</sup> (''a''<sub>''i'',''j''</sub> = ''a''<sub>''j'',''i''</sub>). ||
|-
| [[Toeplitz matrix]] || A matrix with constant diagonals. ||
|-
| [[Totally positive matrix]]  || A matrix with [[determinant]]s 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 of [[two-dimensional array]]
|-
| [[Vandermonde matrix]] || A row consists of 1, ''a'', ''a''<sup>2</sup>, ''a''<sup>3</sup>, 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 (mathematics)|Z-matrix]] || A matrix with all off-diagonal entries less than zero.
|}

== Matrices satisfying some equations ==
A number of matrix-related notions is about properties of products or inverses of the given matrix. The [[matrix product]] of a ''m''-by-''n'' matrix ''A'' and a ''n''-by-''k'' matrix ''B'' is the ''m''-by-''k'' matrix ''C'' given by
:<math> (C)_{i,j} = \sum_{r=1}^n A_{i,r}B_{r,j}.</math><ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|title=Matrix Multiplication|url=https://mathworld.wolfram.com/MatrixMultiplication.html|access-date=2020-09-07|website=mathworld.wolfram.com|language=en}}</ref>
This matrix product is denoted ''AB''. Unlike the product of numbers, matrix products are not [[commutative]], that is to say ''AB'' need not be equal to ''BA''.<ref name=":1" /> A number of notions are concerned with the failure of this commutativity. An [[inverse of a matrix|inverse]] of square matrix ''A'' is a matrix ''B'' (necessarily of the same dimension as ''A'') such that ''AB'' = ''I''. Equivalently, ''BA'' = ''I''. An inverse need not exist. If it exists, ''B'' is uniquely determined, and is also called ''the'' inverse of ''A'', denoted ''A''<sup>&minus;1</sup>.

{| class="wikitable sortable"
! Name !! Explanation !! Notes
|-
|[[Circular matrix]] or [[Coninvolutory matrix]]
|A matrix whose inverse is equal to its entrywise complex conjugate: ''A''<sup>&minus;1</sup> = {{overline|''A''}}.
|Compare with unitary matrices.
|-
|     [[Matrix congruence|Congruent matrix]]  ||  Two matrices ''A'' and ''B'' are congruent if there exists an invertible matrix ''P'' such that {{nowrap|''P''<sup>T</sup> ''A'' ''P''}} = ''B''. || Compare with similar matrices.
|-
|[[EP matrix]] or Range-Hermitian matrix
|A square matrix that commutes with its [[Moore–Penrose inverse]]: ''AA''<sup>+</sup> = ''A''<sup>+</sup>''A.''
|
|-
|     [[Idempotent matrix]] or <br /> [[Projection (linear algebra)|Projection Matrix]]  ||  A matrix that has the property ''A''² = ''AA'' = ''A''. || The name projection matrix inspires from the observation of projection of a point multiple <br /> times onto a subspace(plane or a line) giving the same result as [[Projection (linear algebra)#Properties and classification|one projection]].
|-
|    [[Invertible matrix]]   ||  A square matrix having a multiplicative [[inverse matrix|inverse]], that is, a matrix ''B'' such that ''AB'' = ''BA'' = ''I''. || Invertible matrices form the [[general linear group]].
|-
|   [[Involutory matrix]]    ||  A square matrix which is its own inverse, i.e., ''AA'' = ''I''. || [[Signature matrix|Signature matrices]], [[Householder transformation#Definition and properties|Householder matrices]] (Also known as 'reflection matrices' <br /> to reflect a point about a plane or line) have this property.
|-
|   [[Isometry|Isometric matrix]]    || A matrix that preserves distances, i.e., a matrix that satisfies ''A''<sup>*</sup>''A'' = ''I'' where ''A''<sup>*</sup> denotes the [[conjugate transpose]] of ''A''. || 
|-
|   [[Nilpotent matrix]]    ||  A square matrix satisfying ''A''<sup>''q''</sup> = 0 for some positive integer ''q''. || Equivalently, the only eigenvalue of ''A'' is 0.
|-
| [[Normal matrix]]|| A square matrix that commutes with its [[conjugate transpose]]: ''AA''<sup>∗</sup> = ''A''<sup>∗</sup>''A'' || They are the matrices to which the [[spectral theorem]] applies.
|-
|    [[Orthogonal matrix]]   ||  A matrix whose inverse is equal to its [[transpose]], ''A''<sup>&minus;1</sup> = ''A''<sup>''T''</sup>. || They form the [[orthogonal group]].
|-
| [[Orthonormal matrix]] || A matrix whose columns are [[orthonormal]] vectors. ||
|-
|   [[Partial isometry|Partially Isometric matrix]]    ||  A matrix that is an [[isometry]] on the [[orthogonal complement]] of its [[kernel (algebra)|kernel]]. Equivalently, a matrix that satisfies ''AA''<sup>*</sup>''A'' = ''A''. || Equivalently, a matrix with [[Singular value|singular values]] that are either 0 or 1.
|-
|    [[Singular matrix]]  ||  A square matrix that is not invertible. ||
|-
|   [[Unimodular matrix]] || An invertible matrix with entries in the integers ([[integer matrix]]) || Necessarily the determinant is +1 or &minus;1.
|-
|    [[Unipotent matrix]]   ||  A square matrix with all eigenvalues equal to 1. || Equivalently, {{nowrap|''A'' &minus; ''I''}} is nilpotent. See also [[unipotent group]].
|-
|[[Unitary matrix]]
|A square matrix whose inverse is equal to its [[conjugate transpose]], ''A''<sup>&minus;1</sup> = ''A''<sup>*</sup>.
|
|-
|    [[Totally unimodular matrix]] || A matrix for which every non-singular square submatrix is [[unimodular matrix|unimodular]]. This has some implications in the  [[linear programming]] [[linear programming relaxation|relaxation]] of an [[integer program]]. ||
|-
|     [[Weighing matrix]]  ||  A square matrix the entries of which are in {{nowrap|{{mset|0, 1, &minus;1}}}}, such that ''AA''<sup>T</sup> = ''wI'' for some positive integer ''w''. ||
|}

==Matrices with conditions on eigenvalues or eigenvectors==
{| class="wikitable sortable"
! Name !! Explanation !! Notes
|-
|     [[Convergent matrix]] ||  A square matrix whose successive powers approach the [[zero matrix]]. || Its [[eigenvalues and eigenvectors|eigenvalues]] have magnitude less than one.
|-
|  [[Defective matrix]] || A square matrix that does not have a complete basis of [[eigenvectors]], and is thus not [[Diagonalizable matrix|diagonalizable]].    ||
|-
|[[Derogatory matrix]] 
|A square matrix whose [[Minimal polynomial (linear algebra)|minimal polynomial]] is of order less than ''n''. Equivalently, at least one of its eigenvalues has at least two [[Jordan block]]s.<ref>{{Cite web|title=Non-derogatory matrix - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Non-derogatory_matrix|access-date=2020-09-07|website=encyclopediaofmath.org}}</ref>
|
|-
|  [[Diagonalizable matrix]]     ||   A square matrix [[similar matrix|similar]] to a diagonal matrix. || It has an [[eigenbasis]], that is, a complete set of [[linearly independent]] eigenvectors. 
|-
|   [[Hurwitz-stable matrix|Hurwitz matrix]]    ||  A matrix whose eigenvalues have strictly negative real part. A stable system of differential equations may be represented by a Hurwitz matrix.   ||
|-
|  [[M-matrix]] || A Z-matrix with eigenvalues whose real parts are nonnegative. ||
|-
|     [[Positive-definite matrix]]  ||  A Hermitian matrix with every eigenvalue positive.   ||
|-
|   [[Stability matrix]]  ||  || Synonym for [[Hurwitz-stable matrix|Hurwitz matrix]].
|-
|   [[Stieltjes matrix]] ||  A real symmetric positive definite matrix with nonpositive off-diagonal entries. || Special case of an [[M-matrix]].
|}

== Matrices generated by specific data ==

{| class="wikitable sortable"
! Name !! Definition !! Comments 
|-
| [[Adjugate matrix]] || [[Transpose]] of the [[cofactor matrix]] || The [[inverse matrix|inverse of a matrix]] is its adjugate matrix divided by its [[determinant]]
|-
|  [[Augmented matrix]] ||  Matrix whose rows are concatenations of the rows of two smaller matrices || Used for performing the same [[row operations]] on two matrices
|-
| [[Bézout matrix]] || Square matrix whose [[determinant]] is the [[resultant]] of two polynomials|| See also [[Sylvester matrix]]
|-
|  [[Carleman matrix]] || Infinite matrix of the [[Taylor coefficient]]s of an [[analytic function]] and its integer powers|| The composition of two functions can be expressed as the product of their Carleman matrices
|-
| [[Cartan matrix]] || A matrix associated with either a finite-dimensional [[associative algebra]], or a [[semisimple Lie algebra]] ||
|-
|  [[Cofactor matrix]] || Formed by the [[cofactor (linear algebra)|cofactors]] of a square matrix, that is, the signed [[minor (linear algebra)|minors]], of the matrix || [[Transpose]] of the [[Adjugate matrix]]
|-
|  [[Companion matrix]]  ||  A matrix having the coefficients of a polynomial as last column, and having the polynomial as its [[characteristic polynomial]] ||   
|-
|  [[Coxeter matrix]] || A matrix which describes the relations between the [[involution (mathematics)|involutions]] that generate a [[Coxeter group]]|| 
|-
| [[Distance matrix]]  ||The square matrix formed by the pairwise distances of a set of [[point (geometry)|points]] || [[Euclidean distance matrix]] is a special case 
|-
| [[Euclidean distance matrix]] || A matrix that describes the pairwise distances between [[point (geometry)|points]] in [[Euclidean space]] || See also [[distance matrix]]
|-
|  [[Fundamental matrix (linear differential equation)|Fundamental matrix]] || The matrix formed from the fundamental solutions of a [[system of linear differential equations]] ||
|-
|  [[Generator matrix]] || In [[Coding theory]], a matrix whose rows [[linear span|span]] a [[linear code]] ||
|-
|  [[Gramian matrix]] || The symmetric matrix of the pairwise [[inner product]]s of a set of vectors in an [[inner product space]] ||
|-
| [[Hessian matrix]] || The square matrix of [[Partial derivative|second partial derivatives]] of a [[function of several variables]] ||
|-
|  [[Householder transformation#Householder matrix|Householder matrix]] || The matrix of a [[reflection (mathematics)|reflection]] with respect to a [[hyperplane]] passing through the origin|| 
|-
|  [[Jacobian matrix]] || The matrix of the partial derivatives of a [[function of several variables]] ||
|-
|[[Moment matrix]]
| ||Used in [[statistics]] and [[Sum-of-squares optimization]]
|-
|  [[Payoff matrix]] || A matrix in [[game theory]] and [[economics]], that represents the payoffs in a [[normal form game]] where players move simultaneously || 
|-
|  [[Pick matrix]] || A matrix that occurs in the study of analytical interpolation problems ||
|-
| [[Rotation matrix]] || A matrix representing a [[rotation (geometry)|rotation]]  ||
|-
|  [[Seifert matrix]] || A matrix in [[knot theory]], primarily for the algebraic analysis of topological properties of knots and links.|| [[Alexander polynomial]]
|-
|  [[Shear matrix]]|| The matrix of a [[shear 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 two [[polynomials]] || The Sylvester matrix is nonsingular if and only if the two polynomials are [[coprime]] to each other
|-
| [[Symplectic matrix]]  || The real matrix of a [[symplectic transformation]] ||
|-
| [[Transformation matrix]] || The matrix of a [[linear transformation]] or a [[geometric transformation]]||
|-
| [[Wedderburn matrix]]  || A matrix of the form <math>A - (y^T A x)^{-1} A x y^T A</math>, used for rank-reduction & biconjugate decompositions || Analysis of matrix decompositions
|}

==Matrices used in statistics==
The following matrices find their main application in [[statistics]] and [[probability theory]].
*[[Bernoulli matrix]] —  a square matrix with entries +1, &minus;1, with equal [[probability]] 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 symmetric ''n×n'' matrix, formed by the pairwise [[Pearson product-moment correlation coefficient|correlation coefficient]]s of several [[random variable]]s.
*[[Covariance matrix]] — a symmetric ''n×n'' matrix, formed by the pairwise [[covariance]]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.
*[[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 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.
*[[Stochastic matrix|Transition matrix]] — a matrix representing the [[probabilities]] of conditions changing from one state to another in a [[Markov chain]]
*[[Unistochastic matrix]] — a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some unitary matrix

==Matrices used in graph theory==
The following matrices find their main application in [[graph theory|graph]] and [[network theory]].
*[[Adjacency matrix]] — a square matrix representing a graph, with ''a<sub>ij</sub>'' non-zero if vertex ''i'' and vertex ''j'' are adjacent.
*[[Biadjacency matrix]] — a special class of [[adjacency matrix]] that describes adjacency in [[bipartite graph]]s.
*[[Degree matrix]] — a diagonal matrix defining the degree of each [[vertex (graph theory)|vertex]] in a graph.
*[[Edmonds matrix]] — a square matrix of a bipartite graph.
*[[Incidence matrix]] — a matrix representing a relationship between two classes of objects (usually [[vertex (graph theory)|vertices]] and [[edge (graph theory)|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 &minus;1 for adjacency; +1 for nonadjacency; 0 on the diagonal.
*[[Skew-adjacency matrix]] — an [[adjacency matrix]] in which each non-zero ''a<sub>ij</sub>'' is 1 or &minus;1, accordingly as the direction ''i → j'' matches or opposes that of an initially specified orientation.
*[[Tutte matrix]] — a generalization of the Edmonds matrix for a balanced bipartite graph.

==Matrices used in science and engineering==
*[[Cabibbo–Kobayashi–Maskawa matrix]] — a unitary matrix used in [[particle physics]] to describe the strength of ''flavour-changing'' weak decays.
*[[Density matrix]] — a matrix describing the statistical state of a quantum system. [[Hermitian matrix|Hermitian]], [[non-negative matrix|non-negative]] and with [[trace (linear algebra)|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.
*[[Gamma matrices]] — 4 × 4 matrices in [[quantum field theory]].
*[[Gell-Mann matrices]] — a [[Generalizations of Pauli matrices|generalization of the Pauli matrices]]; these matrices are one notable representation of the [[Lie group#The Lie algebra associated to a Lie group|infinitesimal generator]]s of the [[special unitary group]] SU(3).
*[[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 vector]]s of a [[Quantum mechanics|quantum]] system.
*[[S matrix]] — a matrix in [[quantum 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|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.
*[[Supnick matrix]] — a square matrix used in [[computer science]].
*[[Z-matrix (chemistry)|Z-matrix]] — a matrix in [[chemistry]], representing a molecule in terms of its relative atomic geometry.

==Specific matrices==
*[[Wilson matrix]], a matrix used as an example for test purposes.

==Other matrix-related terms and definitions==

*[[Jordan canonical form]] — an 'almost' diagonalised matrix, where the only non-zero elements appear on the lead and superdiagonals.
*[[Linear independence]] — two or more [[coordinate vector|vectors]] are linearly independent if there is no way to construct one from [[linear combination]]s of the others.
*[[Matrix exponential]] — defined by the [[Exponential function#Formal definition|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)<sup>th</sup> derivative of row one.

==See also==
*[[Perfect matrix]]
{{Portal|Mathematics}}

==Notes==
<references/>

==References==
* {{Citation | last1=Hogben | first1=Leslie|authorlink= Leslie Hogben | title=Handbook of Linear Algebra (Discrete Mathematics and Its Applications) | publisher=Chapman & Hall/CRC | location=Boca Raton | isbn=978-1-58488-510-8 | year=2006}}

{{DEFAULTSORT:Matrices}}
[[Category:Mathematics-related lists]]
[[Category:Matrices (mathematics)| Named]]