List of named matrices

This page 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 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

Several important classes of matrices are subsets of each other.

and the zero matrix of dimension . For example:

.[1]

Further ways of classifying matrices are according to their eigenvalues, or by imposing conditions on the 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.

Matrices with explicitly constrained 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 ai,i. The other diagonal is called anti-diagonal (or counter-diagonal).

NameExplanationNotes, references
(0,1)-matrixA matrix with all elements either 0 or 1.Synonym for binary matrix or logical matrix.
Alternant matrixA matrix in which successive columns have a particular function applied to their entries.
Anti-diagonal matrixA square matrix with all entries off the anti-diagonal equal to zero.
Anti-Hermitian matrixSynonym for skew-Hermitian matrix.
Anti-symmetric matrixSynonym for skew-symmetric matrix.
Arrowhead matrixA square matrix containing zeros in all entries except for the first row, first column, and main diagonal.
Band matrixA square matrix whose non-zero entries are confined to a diagonal band.
Bidiagonal matrixA matrix with elements only on the main diagonal and either the superdiagonal or subdiagonal.Sometimes defined differently, see article.
Binary matrixA matrix whose entries are all either 0 or 1.Synonym for (0,1)-matrix or logical matrix.[2]
Bisymmetric matrixA square matrix that is symmetric with respect to its main diagonal and its main cross-diagonal.
Block-diagonal matrixA block matrix with entries only on the diagonal.
Block matrixA matrix partitioned in sub-matrices called blocks.
Block tridiagonal matrixA block matrix which is essentially a tridiagonal matrix but with submatrices in place of scalar elements.
Boolean matrixA matrix whose entries are taken from a Boolean algebra.
Cauchy matrixA matrix whose elements are of the form 1/(xi + yj) for (xi), (yj) injective sequences (i.e., taking every value only once).
Centrosymmetric matrixA matrix symmetric about its center; i.e., aij = ani+1,nj+1.
Conference matrixA square matrix with zero diagonal and +1 and −1 off the diagonal, such that CTC is a multiple of the identity matrix.
Complex Hadamard matrixA 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 matrixA square matrix A with real coefficients, such that is nonnegative for every nonnegative vector x
Diagonally dominant matrixA matrix whose entries satisfy .
Diagonal matrixA square matrix with all entries outside the main diagonal equal to zero.
Discrete Fourier-transform matrixMultiplying by a vector gives the DFT of the vector as result.
Elementary matrixA square matrix derived by applying an elementary row operation to the identity matrix.
Equivalent matrixA matrix that can be derived from another matrix through a sequence of elementary row or column operations.
Frobenius matrixA square matrix in the form of an identity matrix but with arbitrary entries in one column below the main diagonal.
Generalized permutation matrixA square matrix with precisely one nonzero element in each row and column.
Hadamard matrixA square matrix with entries +1, −1 whose rows are mutually orthogonal.
Hankel matrixA matrix with constant skew-diagonals; also an upside down Toeplitz matrix.A square Hankel matrix is symmetric.
Hermitian matrixA square matrix which is equal to its conjugate transpose, A = A*.
Hessenberg matrixAn "almost" triangular matrix, for example, an upper Hessenberg matrix has zero entries below the first subdiagonal.
Hollow matrixA square matrix whose main diagonal comprises only zero elements.
Integer matrixA matrix whose entries are all integers.
Logical matrixA 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.
Markov matrixA matrix of non-negative real numbers, such that the entries in each row sum to 1.
Metzler matrixA matrix whose off-diagonal entries are non-negative.
Monomial matrixA square matrix with exactly one non-zero entry in each row and column.Synonym for generalized permutation matrix.
Moore matrixA row consists of a, aq, aq², etc., and each row uses a different variable.
Nonnegative matrixA matrix with all nonnegative entries.
Null-symmetric matrix A square matrix whose null space (or kernel) is equal to its transpose, 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 (or kernel) is equal to its conjugate 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 matrixA matrix partitioned into sub-matrices, or equivalently, a matrix whose entries are themselves matrices rather than scalars.Synonym for block matrix.
Parisi matrixA 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 matrixA matrix with the only nonzero entries on the main diagonal and the two diagonals just above and below the main one.
Permutation matrixA matrix representation of a permutation, a square matrix with exactly one 1 in each row and column, and all other elements 0.
Persymmetric matrixA matrix that is symmetric about its northeast-southwest diagonal, i.e., aij = anj+1,ni+1.
Polynomial matrixA matrix whose entries are polynomials.
Positive matrixA matrix with all positive entries.
Quaternionic matrixA matrix whose entries are quaternions.
Sign matrixA matrix whose entries are either +1, 0, or −1.
Signature matrixA diagonal matrix where the diagonal elements are either +1 or −1.
Single-entry matrixA matrix where a single element is one and the rest of the elements are zero.
Skew-Hermitian matrixA square matrix which is equal to the negative of its conjugate transpose, A* = −A.
Skew-symmetric matrixA matrix which is equal to the negative of its transpose, AT = −A.
Skyline matrixA rearrangement of the entries of a banded matrix which requires less space.
Sparse matrixA matrix with relatively few non-zero elements.Sparse matrix algorithms can tackle huge sparse matrices that are utterly impractical for dense matrix algorithms.
Sylvester matrixA 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 matrixA square matrix which is equal to its transpose, A = AT (ai,j = aj,i).
Toeplitz matrixA matrix with constant diagonals.
Triangular matrixA 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 matrixA matrix with the only nonzero entries on the main diagonal and the diagonals just above and below the main one.
Vandermonde matrixA row consists of 1, a, a2, a3, etc., and each row uses a different variable.
Walsh matrixA 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-matrixA 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. The matrix entries will be denoted aij. The table below uses the Kronecker delta δij for two integers i and j which is 1 if i = j and 0 else.

NameExplanationSymbolic description of the entriesNotes
Exchange matrixA binary matrix with ones on the anti-diagonal, and zeroes everywhere else.aij = δn+1−i,jA permutation matrix.
Hilbert matrixaij = (i + j  1)−1.A Hankel matrix.
Identity matrixA square diagonal matrix, with all entries on the main diagonal equal to 1, and the rest 0.aij = δij
Lehmer matrixaij = min(i, j) ÷ max(i, j).A positive symmetric matrix.
Matrix of onesA matrix with all entries equal to one.aij = 1.
Pascal matrixA matrix containing the entries of Pascal's triangle.
Pauli matricesA 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.
Redheffer matrixEncodes a Dirichlet convolution. Matrix entries are given by the divisor function; entires of the inverse are given by the Möbius function.aij are 1 if i divides j or if j = 1; otherwise, aij = 0.A (0, 1)-matrix.
Shift matrixA matrix with ones on the superdiagonal or subdiagonal and zeroes elsewhere.aij = δi+1,j or aij = δi−1,jMultiplication by it shifts matrix elements by one position.
Zero matrixA matrix with all entries equal to zero.aij = 0.

Matrices with conditions on eigenvalues or eigenvectors

NameExplanationNotes
Companion matrixA matrix whose eigenvalues are equal to the roots of the polynomial.
Convergent matrixA square matrix whose successive powers approach the zero matrix.Its eigenvalues have magnitude less than one.
Defective matrixA square matrix that does not have a complete basis of eigenvectors, and is thus not diagonalisable.
Diagonalizable matrixA square matrix similar to a diagonal matrix.It has an eigenbasis, that is, a complete set of linearly independent eigenvectors.
Hurwitz matrixA matrix whose eigenvalues have strictly negative real part. A stable system of differential equations may be represented by a Hurwitz matrix.
M-matrixA Z-matrix with eigenvalues whose real parts are nonnegative.
Positive-definite matrixA Hermitian matrix with every eigenvalue positive.
Stability matrixSynonym for Hurwitz matrix.
Stieltjes matrixA real symmetric positive definite matrix with nonpositive off-diagonal entries.Special case of an M-matrix.

Matrices satisfying conditions on products or inverses

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

[3]

This matrix product is denoted AB.[1] Unlike the product of numbers, matrix products are not commutative, that is to say AB need not be equal to BA.[3] A number of notions are concerned with the failure of this commutativity. An 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 A1.

NameExplanationNotes
Circular matrix or Coninvolutory matrix A matrix whose inverse is equal to its entrywise complex conjugate: A1 = A. Compare with unitary matrices.
Congruent matrixTwo matrices A and B are congruent if there exists an invertible matrix P such that PT A P = B.Compare with similar matrices.
EP matrix or Range-Hermitian matrix A square matrix that commutes with its Moore–Penrose inverse: AA+ = A+A.
Idempotent matrix or
Projection Matrix
A matrix that has the property A² = AA = A.The name projection matrix inspires from the observation of projection of a point multiple
times onto a subspace(plane or a line) giving the same result as one projection.
Invertible matrixA square matrix having a multiplicative inverse, that is, a matrix B such that AB = BA = I.Invertible matrices form the general linear group.
Involutory matrixA square matrix which is its own inverse, i.e., AA = I.Signature matrices, Householder Matrices (Also known as 'reflection matrices'
to reflect a point about a plane or line) have this property.
Nilpotent matrixA square matrix satisfying Aq = 0 for some positive integer q.Equivalently, the only eigenvalue of A is 0.
Normal matrixA square matrix that commutes with its conjugate transpose: AA = AAThey are the matrices to which the spectral theorem applies.
Orthogonal matrixA matrix whose inverse is equal to its transpose, A1 = AT.They form the orthogonal group.
Orthonormal matrixA matrix whose columns are orthonormal vectors.
Singular matrixA square matrix that is not invertible.
Unimodular matrixAn invertible matrix with entries in the integers (integer matrix)Necessarily the determinant is +1 or 1.
Unipotent matrixA square matrix with all eigenvalues equal to 1.Equivalently, A I is nilpotent. See also unipotent group.
Unitary matrix A square matrix whose inverse is equal to its conjugate transpose, A1 = A*.
Totally unimodular matrixA matrix for which every non-singular square submatrix is unimodular. This has some implications in the linear programming relaxation of an integer program.
Weighing matrixA square matrix the entries of which are in {0, 1, 1}, such that AAT = wI for some positive integer w.

Matrices with specific applications

NameExplanationApplicationsNotes
Adjugate matrixThe matrix containing minors of a given square matrix.Calculating inverse matrices via Laplace expansion.
Alternating sign matrixA square matrix of 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.Dodgson condensation to calculate determinants
Augmented matrixA matrix whose rows are concatenations of the rows of two smaller matrices.Calculating inverse matrices.
Bézout matrixA square matrix which may be used as a tool for the efficient location of polynomial zerosControl theory, Stable polynomials
Carleman matrixA matrix that converts composition of functions to multiplication of matrices.
Cartan matrixA matrix associated with a finite-dimensional associative algebra, or a semisimple Lie algebra (the two meanings are distinct).
Circulant matrixA matrix where each row is a circular shift of its predecessor.System of linear equations, discrete Fourier transform
Cofactor matrixA containing the cofactors, i.e., signed minors, of a given matrix.
Commutation matrixA matrix used for transforming the vectorized form of a matrix into the vectorized form of its transpose.
Coxeter matrixA matrix related to Coxeter groups, which describe symmetries in a structure or system.
Derogatory matrix A square matrix whose minimal polynomial is of order less than n. Equivalently, at least one of its eigenvalues has at least two Jordan blocks.[4]
Distance matrixA square matrix containing the distances, taken pairwise, of a set of points.Computer vision, network analysis.See also Euclidean distance matrix.
Duplication matrixA linear transformation matrix used for transforming half-vectorizations of matrices into vectorizations.
Elimination matrixA linear transformation matrix used for transforming vectorizations of matrices into half-vectorizations.
Euclidean distance matrixA matrix that describes the pairwise distances between points in Euclidean space.See also distance matrix.
Fundamental matrix (linear differential equation)A matrix containing the fundamental solutions of a linear ordinary differential equation.
Generator matrixA matrix whose rows generate all elements of a linear code.Coding theory
Gramian matrixA matrix containing the pairwise angles of given vectors in an inner product space.Test linear independence of vectors, including ones in function spaces.They are real symmetric.
Hessian matrixA square matrix of second partial derivatives of a scalar-valued function.Detecting local minima and maxima of scalar-valued functions in several variables; Blob detection (computer vision)
Householder matrixA transformation matrix widely used in matrix algorithms.QR decomposition.
Jacobian matrixA matrix of first-order partial derivatives of a vector-valued function.Implicit function theorem; Smooth morphisms (algebraic geometry).
Moment matrix A symmetric matrix whose elements are the products of common row/column index dependent monomials. Sum-of-squares optimization.
Payoff matrixA matrix in game theory and economics, that represents the payoffs in a normal form game where players move simultaneously
Pick matrixA matrix that occurs in the study of analytical interpolation problems.
Random matrixA matrix whose entries consist of random numbers from some specified random distribution.
Rotation matrixA matrix representing a rotational geometric transformation.Special orthogonal group, Euler angles
Seifert matrixA matrix in knot theory, primarily for the algebraic analysis of topological properties of knots and links.Alexander polynomial
Shear matrixAn elementary matrix whose corresponding geometric transformation is a shear transformation.
Similarity matrixA matrix of scores which express the similarity between two data points.Sequence alignment
Symplectic matrixA square matrix preserving a standard skew-symmetric form.Symplectic group, symplectic manifold.
Totally positive matrixA matrix with determinants of all its square submatrices positive.Generating the reference points of Bézier curve in computer graphics.
Transformation matrixA matrix representing a linear transformation, often from one co-ordinate space to another to facilitate a geometric transform or projection.
Wedderburn matrixA matrix of the form , used for rank-reduction & biconjugate decompositionsAnalysis of matrix decompositions
X-Y-Z matrix A generalisation of the (rectangular) matrix to a cuboidal form (a 3-dimensional array of entries).

Matrices used in statistics

The following matrices find their main application in statistics and probability theory.

  • Bernoulli matrix — a square matrix with entries +1, 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 correlation coefficients of several random variables.
  • Covariance matrix — a symmetric n×n matrix, formed by the pairwise covariances 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.
  • 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 and network theory.

  • Adjacency matrix — a square matrix representing a graph, with aij non-zero if vertex i and vertex j are adjacent.
  • Biadjacency matrix — a special class of adjacency matrix that describes adjacency in bipartite graphs.
  • Degree matrix — a diagonal matrix defining the degree of each 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 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 usual adjacency matrix but with 1 for adjacency; +1 for nonadjacency; 0 on the diagonal.
  • Skew-adjacency matrix — an adjacency matrix in which each non-zero aij is 1 or 1, accordingly as the direction i → j matches or opposes that of an initially specified orientation.
  • Tutte matrix — a generalisation of the Edmonds matrix for a balanced bipartite graph.

Matrices used in science and engineering

See also

Notes

  1. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-09-07.
  2. Hogben 2006,Ch. 31.3.
  3. Weisstein, Eric W. "Matrix Multiplication". mathworld.wolfram.com. Retrieved 2020-09-07.
  4. "Non-derogatory matrix - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2020-09-07.

References

  • 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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