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Leibniz Formula For Determinants
In algebra, the Leibniz formula, named in honor of Gottfried Leibniz, expresses the determinant of a square matrix in terms of permutations of the matrix elements. If A is an n \times n matrix, where a_ is the entry in the i-th row and j-th column of A, the formula is :\det(A) = \sum_ \sgn(\tau) \prod_^n a_ = \sum_ \sgn(\sigma) \prod_^n a_ where \sgn is the sign function of permutations in the permutation group S_n, which returns +1 and -1 for even and odd permutations, respectively. Another common notation used for the formula is in terms of the Levi-Civita symbol and makes use of the Einstein summation notation, where it becomes : \det(A) = \epsilon_ _ \cdots _, which may be more familiar to physicists. Directly evaluating the Leibniz formula from the definition requires \Omega(n! \cdot n) operations in general—that is, a number of operations asymptotically proportional to n factorial—because n! is the number of order-n permutations. This is impractically difficult for eve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication. Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called '' systems of linear equations''. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. Abstract algebra studies algebraic structures, which consist of a set of mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Multiplication Algorithm
Because matrix multiplication is such a central operation in many numerical algorithms, much work has been invested in making matrix multiplication algorithms efficient. Applications of matrix multiplication in computational problems are found in many fields including scientific computing and pattern recognition and in seemingly unrelated problems such as counting the paths through a Graph (graph theory), graph. Many different algorithms have been designed for multiplying matrices on different types of hardware, including parallel computing, parallel and distributed computing, distributed systems, where the computational work is spread over multiple processors (perhaps over a network). Directly applying the mathematical definition of matrix multiplication gives an algorithm that Analysis of algorithms, takes time on the order of Field (mathematics), field operations to multiply two matrices over that field ( in big O notation). Better asymptotic bounds on the time required to mult ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottfried Wilhelm Leibniz
Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many other branches of mathematics, such as binary arithmetic and statistics. Leibniz has been called the "last universal genius" due to his vast expertise across fields, which became a rarity after his lifetime with the coming of the Industrial Revolution and the spread of specialized labor. He is a prominent figure in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history, philology, games, music, and other studies. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in probability theory, biology, medicine, geology, psychology, linguistics and computer science. Leibniz contributed to the field ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinants
In mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . 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. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse. The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries. 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 an matrix can be defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Society For Industrial And Applied Mathematics
Society for Industrial and Applied Mathematics (SIAM) is a professional society dedicated to applied mathematics, computational science, and data science through research, publications, and community. SIAM is the world's largest scientific society devoted to applied mathematics, and roughly two-thirds of its membership resides within the United States. Founded in 1951, the organization began holding annual national meetings in 1954, and now hosts conferences, publishes books and scholarly journals, and engages in advocacy in issues of interest to its membership. Members include engineers, scientists, and mathematicians, both those employed in academia and those working in industry. The society supports educational institutions promoting applied mathematics. SIAM is one of the four member organizations of the Joint Policy Board for Mathematics. Membership Membership is open to both individuals and organizations. By the end of its first full year of operation, SIAM had 130 me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cramer's Rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-sides of the equations. It is named after Gabriel Cramer, who published the rule for an arbitrary number of unknowns in 1750, although Colin Maclaurin also published special cases of the rule in 1748, and possibly knew of it as early as 1729. Cramer's rule, implemented in a naive way, is computationally inefficient for systems of more than two or three equations. In the case of equations in unknowns, it requires computation of determinants, while Gaussian elimination produces the result with the same (up to a constant factor independent of ) computational complexity as the computation of a single determinant. Moreo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Expansion
In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an -matrix as a weighted sum of minors, which are the determinants of some - submatrices of . Specifically, for every , the ''Laplace expansion along the th row'' is the equality \begin \det(B)&= \sum_^ (-1)^ b_ m_, \end where b_ is the entry of the th row and th column of , and m_ is the determinant of the submatrix obtained by removing the th row and the th column of . Similarly, the ''Laplace expansion along the th column'' is the equality \begin \det(B)&= \sum_^ (-1)^ b_ m_. \end (Each identity implies the other, since the determinants of a matrix and its transpose are the same.) The coefficient (-1)^ m_ of b_ in the above sum is called the cofactor of b_ in . The Laplace expansion is often useful in proofs, as in, for example, allowing recursion on the size of matrices. It is also of didactic interest for its simplicity an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (: matrices) is a rectangle, rectangular array or table of numbers, symbol (formal), symbols, or expression (mathematics), expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a " matrix", or a matrix of dimension . Matrices are commonly used in linear algebra, where they represent linear maps. In geometry, matrices are widely used for specifying and representing geometric transformations (for example rotation (mathematics), rotations) and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Map
In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (for example, a bilinear form or a multilinear form) that is zero whenever any pair of its arguments is equal. This generalizes directly to a module over a commutative ring. The notion of alternatization (or alternatisation) is used to derive an alternating multilinear map from any multilinear map of which all arguments belong to the same space. Definition Let R be a commutative ring and , W be modules over R. A multilinear map of the form f: V^n \to W is said to be alternating if it satisfies the following equivalent conditions: # whenever there exists 1 \leq i \leq n-1 such that x_i = x_ then . # whenever there exists 1 \leq i \neq j \leq n such that x_i = x_j then . Vector spaces Let V, W be vector spaces over the same field. Then a multilinear map of the form f: V^n \to W is alternating if it satisfies the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multilinear Map
Multilinear may refer to: * Multilinear form, a type of mathematical function from a vector space to the underlying field * Multilinear map, a type of mathematical function between vector spaces * Multilinear algebra, a field of mathematics {{mathdab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternatization
In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (for example, a bilinear form or a multilinear form) that is zero whenever any pair of its arguments is equal. This generalizes directly to a module over a commutative ring. The notion of alternatization (or alternatisation) is used to derive an alternating multilinear map from any multilinear map of which all arguments belong to the same space. Definition Let R be a commutative ring and , W be modules over R. A multilinear map of the form f: V^n \to W is said to be alternating if it satisfies the following equivalent conditions: # whenever there exists 1 \leq i \leq n-1 such that x_i = x_ then . # whenever there exists 1 \leq i \neq j \leq n such that x_i = x_j then . Vector spaces Let V, W be vector spaces over the same field. Then a multilinear map of the form f: V^n \to W is alternating if it satisfies the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
