Related Research Articles
In mathematics, the determinant is a scalar-valued 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.
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse of an invertible matrix. The method is named after Carl Friedrich Gauss (1777–1855). To perform row reduction on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible. There are three types of elementary row operations:
In linear algebra, the rank of a matrix A is the dimension of the vector space generated by its columns. This corresponds to the maximal number of linearly independent columns of A. This, in turn, is identical to the dimension of the vector space spanned by its rows. Rank is thus a measure of the "nondegenerateness" of the system of linear equations and linear transformation encoded by A. There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics.
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with the identity matrix as the identity element of the group. The group is so named because the columns of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.
In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n × n orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact.
In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows in a Hadamard matrix represents two perpendicular vectors, while in combinatorial terms, it means that each pair of rows has matching entries in exactly half of their columns and mismatched entries in the remaining columns. It is a consequence of this definition that the corresponding properties hold for columns as well as rows.
In mathematics, a unimodular matrixM is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix N that is its inverse. Thus every equation Mx = b, where M and b both have integer components and M is unimodular, has an integer solution. The n × n unimodular matrices form a group called the n × n general linear group over , which is denoted .
Sylver coinage is a mathematical game for two players, invented by John H. Conway. The two players take turns naming positive integers that are not the sum of nonnegative multiples of previously named integers. The player who names 1 loses. For instance, if player A opens with 2, B can win by naming 3 as A is forced to name 1. Sylver coinage is an example of a game using misère play because the player who is last able to move loses.
The Sylvester–Gallai theorem in geometry states that every finite set of points in the Euclidean plane has a line that passes through exactly two of the points or a line that passes through all of them. It is named after James Joseph Sylvester, who posed it as a problem in 1893, and Tibor Gallai, who published one of the first proofs of this theorem in 1944.
In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix, and the adjacency matrices used in graph theory, amongst many others. Integer matrices find frequent application in combinatorics.
In mathematics, a regular matroid is a matroid that can be represented over all fields.
Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes.
In mathematics, a matrix is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object.
Sylvester's theorem or the Sylvester theorem may refer to any of several theorems named after James Joseph Sylvester:
Leonid Mirsky was a Russian-British mathematician who worked in number theory, linear algebra, and combinatorics. Mirsky's theorem is named after him.
In geometry, a Sylvester–Gallai configuration consists of a finite subset of the points of a projective space with the property that the line through any two of the points in the subset also passes through at least one other point of the subset.
In matroid theory, a Sylvester matroid is a matroid in which every pair of elements belongs to a three-element circuit of the matroid.
In linear algebra, a branch of mathematics, a (multiplicative) compound matrix is a matrix whose entries are all minors, of a given size, of another matrix. Compound matrices are closely related to exterior algebras, and their computation appears in a wide array of problems, such as in the analysis of nonlinear time-varying dynamical systems and generalizations of positive systems, cooperative systems and contracting systems.
References
- ↑ Franklin, Fabian (1897), "James Joseph Sylvester", Bulletin of the American Mathematical Society, 3 (9): 299–309, doi: 10.1090/S0002-9904-1897-00424-4 , MR 1557527 .
- ↑ MathSciNet lists over 500 mathematics articles with "Sylvester" in their titles, most of which concern mathematical subjects named after Sylvester.
- ↑ Borwein, P.; Moser, W. O. J. (1990), "A survey of Sylvester's problem and its generalizations", Aequationes Mathematicae , 40 (1): 111–135, CiteSeerX 10.1.1.218.8616 , doi:10.1007/BF02112289, S2CID 122052678 .
- ↑ Murty, U. S. R. (1969), "Sylvester matroids", Recent Progress in Combinatorics (Proc. Third Waterloo Conf. on Combinatorics, 1968), New York: Academic Press, pp. 283–286, MR 0255432 .
- ↑ Erwin H. Bareiss (1968), Sylvester's Identity and Multistep Integer- Preserving Gaussian Elimination. Mathematics of Computation, Vol. 22, No. 103, pp. 565–578
- ↑ Berlekamp, Elwyn R.; Conway, John H.; Guy, Richard K. (1982), "Sylver Coinage", Winning Ways for your Mathematical Plays, Vol. 2: Games in Particular, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], pp. 576, 606, MR 0654502 .
- ↑ Cantor, Geoffrey (2004), "Creating the Royal Society's Sylvester Medal" (PDF), British Journal for the History of Science, 37 (1(132)): 75–92, doi:10.1017/S0007087403005132, MR 2128208, S2CID 143307164