Matrices With Specific Applications
| Name | Explanation | Used in | Notes |
|---|---|---|---|
| Adjugate matrix | The matrix containing minors of a given square matrix. | Calculating inverse matrices via Laplace's formula. | |
| Alternating sign matrix | A 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 matrix | A matrix whose rows are concatenations of the rows of two smaller matrices. | Calculating inverse matrices. | |
| Bézout matrix | A square matrix which may be used as a tool for the efficient location of polynomial zeros | Control theory, Stable polynomials | |
| Carleman matrix | A matrix that converts composition of functions to multiplication of matrices. | ||
| Cartan matrix | A matrix associated with a finite-dimensional associative algebra, or a semisimple Lie algebra (the two meanings are distinct). | ||
| Circulant matrix | A matrix where each row is a circular shift of its predecessor. | System of linear equations, discrete Fourier transform | |
| Cofactor matrix | A containing the cofactors, i.e., signed minors, of a given matrix. | ||
| Commutation matrix | A matrix used for transforming the vectorized form of a matrix into the vectorized form of its transpose. | ||
| Coxeter matrix | A matrix related to Coxeter groups, which describe symmetries in a structure or system. | ||
| Distance matrix | A square matrix containing the distances, taken pairwise, of a set of points. | Computer vision, network analysis. | See also Euclidean distance matrix. |
| Duplication matrix | A linear transformation matrix used for transforming half-vectorizations of matrices into vectorizations. | ||
| Elimination matrix | A linear transformation matrix used for transforming vectorizations of matrices into half-vectorizations. | ||
| Euclidean distance matrix | A 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 matrix | A matrix whose rows generate all elements of a linear code. | Coding theory | |
| Gramian matrix | A 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 matrix | A 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 matrix | A transformation matrix widely used in matrix algorithms. | QR decomposition. | |
| Jacobian matrix | A matrix of first-order partial derivatives of a vector-valued function. | Implicit function theorem; Smooth morphisms (algebraic geometry). | |
| 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. | ||
| Random matrix | A matrix whose entries consist of random numbers from some specified random distribution. | ||
| Rotation matrix | A matrix representing a rotational geometric transformation. | Special orthogonal group, Euler angles | |
| Seifert matrix | A matrix in knot theory, primarily for the algebraic analysis of topological properties of knots and links. | Alexander polynomial | |
| Shear matrix | An elementary matrix whose corresponding geometric transformation is a shear transformation. | ||
| Similarity matrix | A matrix of scores which express the similarity between two data points. | Sequence alignment | |
| Symplectic matrix | A square matrix preserving a standard skew-symmetric form. | Symplectic group, symplectic manifold. | |
| Totally positive matrix | A matrix with determinants of all its square submatrices positive. | Generating the reference points of Bézier curve in computer graphics. | |
| Transformation matrix | A matrix representing a linear transformation, often from one co-ordinate space to another to facilitate a geometric transform or projection. | ||
| Wedderburn matrix | A matrix of the form, used for rank-reduction & biconjugate decompositions | Analysis of matrix decompositions |
- Derogatory matrix — a square n×n matrix whose minimal polynomial is of order less than n.
- Moment matrix — a symmetric matrix whose elements are the products of common row/column index dependent monomials.
- X-Y-Z matrix — a generalisation of the (rectangular) matrix to a cuboidal form (a 3-dimensional array of entries).
Read more about this topic: List Of Matrices
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