This page lists some important classes of matrices
used in mathematics
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 of permutation matrices that arises from Dodgson condensation.
- Anti-diagonal matrix — a square matrix with all entries off the anti-diagonal equal to zero.
- Anti-Hermitian matrix — another name for a skew-Hermitian matrix.
- Anti-symmetric matrix — another name for a skew-symmetric matrix.
- Arrowhead matrix — a square matrix containing zeros in all entries except for the first row, first column, and main diagonal
- Augmented matrix — a matrix whose rows are concatenations of the rows of two smaller matrices.
- Band matrix — a square matrix whose non-zero entries are confined to a diagonal band.
- Bézout matrix — a square matrix which may be used as a tool for the efficient location of polynomial zeros
- Bidiagonal matrix — a matrix with elements only on the main diagonal and either the superdiagonal or subdiagonal (sometimes defined differently - see article).
- Binary matrix — another name for a (0,1)-matrix (a matrix whose coefficients are all either 0 or 1).
- Bisymmetric matrix — a square matrix that is symmetric with respect to its main diagonal and its main cross-diagonal.
- Block-diagonal matrix — a block matrix with entries only on the diagonal.
- Block matrix — a matrix partitioned in sub-matrices called blocks.
- Block tridiagonal matrix — a block matrix which is essentially a tridiagonal matrix but with submatrices in place of scalar elements.
- Carleman matrix — a matrix that converts composition of functions to multiplication of matrices
- Cartan matrix — a matrix representing a non-semisimple finite-dimensional algebra, or a Lie algebra (note that the two are distinct)
- Cauchy matrix — a matrix whose elements are of the form 1/(xi + yj) for (xi), (yj) injective sequences.
- 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.
- 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 CTC 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 PT A P = B
- Copositive matrix — a square matrix A with real coefficients, such that is nonnegative for every nonnegative vector x
- Coxeter matrix — a matrix related to Coxeter groups, which describe symmetries in a structure or system.
- Defective matrix — a square matrix that does not have a complete basis of eigenvectors, and is thus not diagonalisable.
- Derogatory matrix — a square n×n matrix whose minimal polynomial is of order less than n.
- Diagonally dominant matrix — a matrix whose entries satisfy |aii| > Σj≠i |aij|.
- Diagonal matrix — a square matrix with all entries off the main diagonal equal to zero.
- Diagonalizable matrix — a square matrix similar to a diagonal matrix. It has a complete set of linearly independent eigenvectors.
- Distance matrix — a square matrix containing the distances, taken pairwise, of a set of points.
- Duplication matrix — a linear transformation matrix used for transforming half-vectorizations of matrices into vectorizations.
- Elementary matrix — a matrix derived by applying an elementary row operation to the identity matrix.
- Elimination matrix — a linear transformation matrix used for transforming vectorizations of matrices into half-vectorizations.
- Equivalent matrix — a matrix that can be derived from another matrix through a sequence of elementary row or column operations.
- Euclidean distance matrix — a matrix that describes the pairwise distances between points in Euclidean space.
- Frobenius matrix — a matrix in the form of an identity matrix but with arbitrary entries in one column below the main diagonal
- Fundamental matrix (linear differential equation)
- Gell-Mann matrices — a generalisation of the Pauli matrices, these matrices are one notable representation of the infinitesimal generators of the special unitary group, SU(3).
- Generalized permutation matrix — a square matrix with precisely one nonzero element in each row and column.
- Generator matrix — a matrix in coding theory whose rows generate all elements of a linear code.
- Gramian matrix — a real symmetric matrix that can be used to test for linear independence of any function.
- Hadamard matrix — a square matrix with entries +1, −1 whose rows are mutually orthogonal.
- Hankel matrix — a matrix with constant skew-diagonals; also an upside down Toeplitz matrix. A square Hankel matrix is symmetric.
- Hat matrix - a square matrix used in statistics to relate fitted values to observed values.
- Hermitian matrix — a square matrix which is equal to its conjugate transpose, A = A*.
- Hessenberg matrix — an "almost" triangular matrix, for example, an upper Hessenberg matrix has zero entries below the first subdiagonal.
- Hessian matrix — a square matrix of second partial derivatives of a scalar-valued function.
- Hollow matrix — a square matrix whose diagonal comprises only zero elements.
- Householder matrix — a transformation matrix widely used in matrix algorithms.
- Hurwitz matrix — a matrix whose eigenvalues have strictly negative real part. A stable system of differential equations may be represented by a Hurwitz matrix.
- Idempotent matrix — a matrix that has the property A² = AA = A.
- Incidence matrix — a matrix representing a relationship between two classes of objects (used both inside and outside of graph theory).
- Integer matrix — a matrix whose elements are all integers.
- Invertible matrix — a square matrix with a multiplicative inverse.
- Involutary matrix — any square matrix which is its own inverse, such as a signature matrix
- Jacobian matrix — a matrix of first-order partial derivatives of a vector-valued function.
- Logical matrix — a k-dimensional array of boolean values that represents a k-adic relation.
- Moment matrix — a symmetric matrix whose elements are the products of common row/column index dependent monomials.
- Monomial matrix — a square matrix with exactly one non-zero entry in each row and column. Another name for generalized permutation matrix.
- Moore matrix — a row consists of 1, a, aq, aq², etc., and each row uses a different variable
- Nilpotent matrix — a square matrix M such that Mq = 0 for some positive integer q.
- Nonnegative matrix — a matrix with all nonnegative entries.
- Normal matrix — a square matrix that commutes with its conjugate transpose. Normal matrices are precisely the matrices to which the spectral theorem applies.
- Orthogonal matrix — a matrix whose inverse is equal to its transpose, A−1 = AT.
- Orthonormal matrix — a matrix whose columns are orthonormal vectors.
- Partitioned matrix — another name for a block matrix (a matrix partitioned into sub-matrices, or equivalently, a matrix whose elements are themselves matrices rather than scalars)
- Payoff matrix — a matrix in game theory, that represents the payoffs in a normal form game where players move simultaneously
- Pentadiagonal matrix — a matrix with the only nonzero entries on the main diagonal and the two diagonals just above and below the main one.
- Permutation matrix — a matrix representation of a permutation, a square matrix with exactly one 1 in each row and column, and all other elements 0.
- Persymmetric matrix — a matrix that is symmetric about its northeast-southwest diagonal, i.e., aij = an−j+1,n−i+1
- Pick matrix — a matrix that occurs in the study of analytical interpolation problems.
- Polynomial matrix — a matrix with polynomials as its elements.
- Positive-definite matrix — a Hermitian matrix with every eigenvalue positive.
- Positive matrix — a matrix with all positive entries.
- Random matrix — a matrix of given type and size whose entries consist of random numbers from some specified distribution.
- Rotation matrix — a matrix representing a rotational geometric transformation.
- Seifert matrix — a matrix in knot theory, primarily for the algebraic analysis of topological properties of knots and links.
- Shear matrix — an elementary matrix whose corresponding geometric transformation is a shear transformation.
- Sign matrix — a matrix whose elements are either +1, 0, or −1.
- Signature matrix — a diagonal matrix where the diagonal elements are either +1 or −1.
- Similar matrix — two matrices A and B are called similar if there exists an invertible matrix P such that P−1AP = B.
- Similarity matrix — a matrix of scores which express the similarity between two data points.
- Singular matrix — a noninvertible square matrix.
- Skew-Hermitian matrix — a square matrix which is equal to the negative of its conjugate transpose, A* = −A.
- Skew-symmetric matrix — a matrix which is equal to the negative of its transpose, 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.
- Square matrix — an n by n matrix. The set of all square matrices form an associative algebra with identity.
- Stability matrix — another name for a Hurwitz matrix.
- Stieltjes matrix — an M-matrix which is symmetric and has an inverse.
- Sylvester matrix — a square matrix whose entries come from coefficients of two polynomials. The Sylvester matrix is nonsingular if and only if the two polynomials are coprime to each other.
- Symmetric matrix — a square matrix which is equal to its transpose, A = AT (ai,j = aj,i ).
- Symplectic matrix — a square matrix preserving a standard skew-symmetric form.
- Toeplitz matrix — a matrix with constant diagonals.
- Totally positive matrix — a matrix with determinants of all its square submatrices positive. It is used in generating the reference points of Bézier curve in computer graphics.
- Totally unimodular matrix — a matrix for which every non-singular square submatrix is unimodular. This has some implications in the linear programming relaxation of an integer program.
- Transformation matrix — a matrix representing a linear transformation, often from one co-ordinate space to another to facilitate a geometric transform or projection.
- Triangular matrix — a matrix with all entries above the main diagonal equal to zero (lower triangular) or with all entries below the main diagonal equal to zero (upper triangular).
- Tridiagonal matrix — a matrix with the only nonzero entries on the main diagonal and the diagonals just above and below the main one.
- Unimodular matrix — a square matrix with determinant +1 or −1.
- Unipotent matrix — a square matrix with all eigenvalues equal to 1.
- Unitary matrix — a square matrix whose inverse is equal to its conjugate transpose, A−1 = A*.
- Vandermonde matrix — a row consists of 1, a, a², a³, etc., and each row uses a different variable
- Weighing matrix — a square matrix , the entries of which are in , such that for some positive integer .
- Walsh matrix — a square matrix, with dimensions a power of 2, the entries of which are +1 or -1.
- X-Y-Z matrix — a generalisation of the (rectangular) matrix to a cuboidal form (a 3-dimensional array of entries).
- Z-matrix — a matrix with all off-diagonal entries less than zero.
The list below comprises matrices whose elements are constant for any given dimension (size) of matrix.
- Exchange matrix — a binary matrix with ones on the anti-diagonal, and zeroes everywhere else.
- Hilbert matrix — a Hankel matrix with elements Hij = (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 aij are given by min(i,j) ÷ max(i,j).
- Pascal matrix — a matrix containing the entries of Pascal's triangle.
- Pauli matrices — a set of three 2 × 2 complex Hermitian and unitary matrices. When combined with the I2 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
and probability theory
Matrices used in graph theory
The following matrices find their main application in graph
and network theory
Matrices used in science and engineering
Other matrix-related terms and definitions