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Dodgson Condensation
In mathematics, Dodgson condensation or method of contractants is a method of computing the determinants of square matrix, square matrices. It is named for its inventor, Charles Lutwidge Dodgson (better known by his pseudonym, as Lewis Carroll, the popular author). The method in the case of an ''n'' × ''n'' matrix is to construct an (''n'' − 1) × (''n'' − 1) matrix, an (''n'' − 2) × (''n'' − 2), and so on, finishing with a 1 × 1 matrix, which has one entry, the determinant of the original matrix. General method This algorithm can be described in the following four steps: # Let A be the given ''n'' × ''n'' matrix. Arrange A so that no zeros occur in its interior. An explicit definition of interior would be all ai,j with i,j\ne1,n. One can do this using any operation that one could normally perform without changing the value of the determinant, such as adding a multiple of one row t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix is denoted , , or . The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e & f \\ g & h & i \end= aei + bfg + cdh - ceg - bdi - afh. The determinant of a matrix can be defined in several equivalent ways. Leibniz formula expresses the determinant as a sum of signed products of matrix entries such that each summand is the product of different entries, and the number of these summands is n!, the factorial of (t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if R is a square matrix representing a rotation (rotation matrix) and \mathbf is a column vector describing the position of a point in space, the product R\mathbf yields another column vector describing the position of that point after that rotation. If \mathbf is a row vector, the same transformation can be obtained using where R^ is the transpose of Main diagonal The entries a_ (''i'' = 1, …, ''n'') form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements , , , . The d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Lutwidge Dodgson
Charles Lutwidge Dodgson (; 27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet and mathematician. His most notable works are ''Alice's Adventures in Wonderland'' (1865) and its sequel ''Through the Looking-Glass'' (1871). He was noted for his facility with word play, logic, and fantasy. His poems ''Jabberwocky'' (1871) and ''The Hunting of the Snark'' (1876) are classified in the genre of literary nonsense. Carroll came from a family of high-church Anglicans, and developed a long relationship with Christ Church, Oxford, where he lived for most of his life as a scholar and teacher. Alice Liddell, the daughter of Christ Church's dean Henry Liddell, is widely identified as the original inspiration for ''Alice in Wonderland'', though Carroll always denied this. An avid puzzler, Carroll created the word ladder puzzle (which he then called "Doublets"), which he published in his weekly column for '' Vanity Fair'' magazine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylvester's Determinant Identity
In matrix theory, Sylvester's determinant identity is an identity useful for evaluating certain types of determinants. It is named after James Joseph Sylvester, who stated this identity without proof in 1851. Cited in Given an ''n''-by-''n'' matrix A, let \det(A) denote its determinant. Choose a pair :u =(u_1, \dots, u_m), v =(v_1, \dots, v_m) \subset (1, \dots, n) of ''m''-element ordered subsets of (1, \dots, n), where ''m'' ≤ ''n''. Let A^u_v denote the (''n''−''m'')-by-(''n''−''m'') submatrix of A obtained by deleting the rows in u and the columns in v. Define the auxiliary ''m''-by-''m'' matrix \tilde^u_v whose elements are equal to the following determinants : (\tilde^u_v)_ := \det(A^_), where uhat/math>, vhat/math> denote the ''m''−1 element subsets of u and v obtained by deleting the elements u_i and v_j, respectively. Then the following is Sylvester's determinantal identity (Sylvester, 1851): :\det(A)(\det(A^u_v))^=\det(\tilde^u_v). When ''m'' =& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjugate Matrix
In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix and is denoted by . It is also occasionally known as adjunct matrix, or "adjoint", though the latter today normally refers to a different concept, the adjoint operator which is the conjugate transpose of the matrix. The product of a matrix with its adjugate gives a diagonal matrix (entries not on the main diagonal are zero) whose diagonal entries are the determinant of the original matrix: :\mathbf \operatorname(\mathbf) = \det(\mathbf) \mathbf, where is the identity matrix of the same size as . Consequently, the multiplicative inverse of an invertible matrix can be found by dividing its adjugate by its determinant. Definition The adjugate of is the transpose of the cofactor matrix of , :\operatorname(\mathbf) = \mathbf^\mathsf. In more detail, suppose is a unital commutative ring and is an matrix with entries from . The -''minor'' of , denoted , is the determ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Bressoud
David Marius Bressoud (born March 27, 1950 in Bethlehem, Pennsylvania) is an American mathematician who works in number theory, combinatorics, and special functions. As of 2019 he is DeWitt Wallace Professor of Mathematics at Macalester College, Director of the Conference Board of the Mathematical Sciences and a former President of the Mathematical Association of America. Life and education Bressoud was born March 27, 1950 in Bethlehem, Pennsylvania. He became interested in mathematics in the seventh grade, where he had a teacher who encouraged him and gave him challenging problems. He attended Albert Wilansky's National Science Foundation summer program at Lehigh University between his junior and senior years in high school, where he also spent most of his time working on problems. He graduated from Swarthmore College in 1971. When he started at Swarthmore he had not yet decided on a major, but after his first year he decided to get out of college as quickly as possibly and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Combinatorial Theory
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. An electronic, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doron Zeilberger
Doron Zeilberger (דורון ציילברגר, born 2 July 1950 in Haifa, Israel) is an Israeli mathematician, known for his work in combinatorics. Education and career He received his doctorate from the Weizmann Institute of Science in 1976, under the direction of Harry Dym, with the thesis "New Approaches and Results in the Theory of Discrete Analytic Functions." He is a Board of Governors Professor of Mathematics at Rutgers University. Contributions Zeilberger has made contributions to combinatorics, hypergeometric identities, and q-series. Zeilberger gave the first proof of the alternating sign matrix conjecture, noteworthy not only for its mathematical content, but also for the fact that Zeilberger recruited nearly a hundred volunteer checkers to "pre-referee" the paper. In 2011, together with Manuel Kauers and Christoph Koutschan, Zeilberger proved the ''q''-TSPP conjecture, which was independently stated in 1983 by George Andrews and David P. Robbins. Zeilberger is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinants
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix is denoted , , or . The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e & f \\ g & h & i \end= aei + bfg + cdh - ceg - bdi - afh. The determinant of a matrix can be defined in several equivalent ways. Leibniz formula expresses the determinant as a sum of signed products of matrix entries such that each summand is the product of different entries, and the number of these summands is n!, the factorial of (the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
