In mathematics, the determinant is a scalar value that is a certain function of the entries of a square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the corresponding linear map is an isomorphism. The determinant of a product of matrices is the product of their determinants.
In linear algebra, the trace of a square matrix A, denoted tr(A), is defined to be the sum of elements on the main diagonal of A. The trace is only defined for a square matrix.
In mathematics, a Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j:
In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix. Both are special cases of a more general function of a matrix called the immanant.
In linear algebra, an n-by-n square matrix A is called invertible if there exists an n-by-n square matrix B such that
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column with all other entries 0. An n × n permutation matrix can represent a permutation of n elements. Pre-multiplying an n-row matrix M by a permutation matrix P, forming PM, results in permuting the rows of M, while post-multiplying an n-column matrix M, forming MP, permutes the columns of M.
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero.
Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a data set. MDS is used to translate distances between each pair of objects in a set into a configuration of points mapped into an abstract Cartesian space.
In mathematics, the determinant of an m×m skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries, a polynomial with integer coefficients that only depends on m. When m is odd, the polynomial is zero. When m is even, it is a nonzero polynomial of degree m/2, and is unique up to multiplication by ±1. The convention on skew-symmetric tridiagonal matrices, given below in the examples, then determines one specific polynomial, called the Pfaffian polynomial. The value of this polynomial, when applied to the entries of a skew-symmetric matrix, is called the Pfaffian of that matrix. The term Pfaffian was introduced by Cayley, who indirectly named them after Johann Friedrich Pfaff.
In matrix theory, the Perron–Frobenius theorem, proved by Oskar Perron and Georg Frobenius, asserts that a real square matrix with positive entries has a unique eigenvalue of largest magnitude and that eigenvalue is real. The corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. This theorem has important applications to probability theory ; to the theory of dynamical systems ; to economics ; to demography ; to social networks ; to Internet search engines (PageRank); and even to ranking of American football teams. The first to discuss the ordering of players within tournaments using Perron–Frobenius eigenvectors is Edmund Landau.
In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also interact with matrix multiplication.
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled randomly from a probability distribution. Random matrix theory (RMT) is the study of properties of random matrices, often as they become large. RMT provides techniques like mean-field theory, diagrammatic methods, the cavity method, or the replica method to compute quantities like traces, spectral densities, or scalar products between eigenvectors. Many physical phenomena, such as the spectrum of nuclei of heavy atoms, the thermal conductivity of a lattice, or the emergence of quantum chaos, can be modeled mathematically as problems concerning large, random matrices.
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A.
In mathematics, especially in probability and combinatorics, a doubly stochastic matrix (also called bistochastic matrix) is a square matrix of nonnegative real numbers, each of whose rows and columns sums to 1, i.e.,
In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix. The LU decomposition was introduced by the Polish astronomer Tadeusz Banachiewicz in 1938. To quote: "It appears that Gauss and Doolittle applied the method [of elimination] only to symmetric equations. More recent authors, for example, Aitken, Banachiewicz, Dwyer, and Crout … have emphasized the use of the method, or variations of it, in connection with non-symmetric problems … Banachiewicz … saw the point … that the basic problem is really one of matrix factorization, or “decomposition” as he called it." It is also sometimes referred to as LR decomposition.
In mathematics, a unistochastic matrix is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some unitary matrix.
The block Wiedemann algorithm for computing kernel vectors of a matrix over a finite field is a generalization by Don Coppersmith of an algorithm due to Doug Wiedemann.
In linear algebra, the computation of the permanent of a matrix is a problem that is thought to be more difficult than the computation of the determinant of a matrix despite the apparent similarity of the definitions.
The iterative proportional fitting procedure is the operation of finding the fitted matrix which is the closest to an initial matrix but with the row and column totals of a target matrix . The fitted matrix being of the form , where and are diagonal matrices such that has the margins of . Some algorithms can be chosen to perform biproportion. We have also the entropy maximization, information loss minimization or RAS which consists of factoring the matrix rows to match the specified row totals, then factoring its columns to match the specified column totals; each step usually disturbs the previous step's match, so these steps are repeated in cycles, re-adjusting the rows and columns in turn, until all specified marginal totals are satisfactorily approximated. However, all algorithms give the same solution. In three- or more-dimensional cases, adjustment steps are applied for the marginals of each dimension in turn, the steps likewise repeated in cycles.
Matrix completion is the task of filling in the missing entries of a partially observed matrix, which is equivalent to performing data imputation in statistics. A wide range of datasets are naturally organized in matrix form. One example is the movie-ratings matrix, as appears in the Netflix problem: Given a ratings matrix in which each entry represents the rating of movie by customer , if customer has watched movie and is otherwise missing, we would like to predict the remaining entries in order to make good recommendations to customers on what to watch next. Another example is the document-term matrix: The frequencies of words used in a collection of documents can be represented as a matrix, where each entry corresponds to the number of times the associated term appears in the indicated document.
