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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. 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 A−1. Name Explanation Notes Congruent matrix Two matrices A and B are congruent if there exists an invertible matrix P such that PT A P = B. Compare with similar matrices. 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 matrix A square matrix having a multiplicative inverse, that is, a matrix B such that AB = BA = I. Invertible matrices form the general linear group. Involutary matrix A 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 matrix A square matrix satisfying Aq = 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∗ = A∗A They are the matrices to which the spectral theorem applies. Orthogonal matrix A matrix whose inverse is equal to its transpose, A−1 = AT. They form the orthogonal group. Orthonormal matrix A matrix whose columns are orthonormal vectors. 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 −1. Unipotent matrix A square matrix with all eigenvalues equal to 1. Equivalently, A − I is nilpotent. See also unipotent group. 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. Weighing matrix A square matrix the entries of which are in {0, 1, −1}, such that AAT = wI for some positive integer w. Matrices with specific applications[edit] 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 A - (y^T A x)^{-1} A x y^T A, 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). |
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