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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 {\displaystyle A} is an n × n {\displaystyle n\times n} matrix, where a i j {\displaystyle a_{ij}} is the entry in the i {\displaystyle i} -th row and j {\displaystyle j} -th ...
Let A be an m × n matrix and k an integer with 0 < k ≤ m, and k ≤ n.A k × k minor of A, also called minor determinant of order k of A or, if m = n, the (n − k) th minor determinant of A (the word "determinant" is often omitted, and the word "degree" is sometimes used instead of "order") is the determinant of a k × k matrix obtained from A by deleting m − k rows and n − k columns.
During execution of the Bareiss algorithm, every integer that is computed is the determinant of a submatrix of the input matrix. This allows, using the Hadamard inequality, to bound the size of these integers. Otherwise, the Bareiss algorithm may be viewed as a variant of Gaussian elimination and needs roughly the same number of arithmetic ...
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. [1]If A is a differentiable map from the real numbers to n × n matrices, then
The optimal number of field operations needed to multiply two square n × n matrices up to constant factors is still unknown. This is a major open question in theoretical computer science . As of January 2024 [update] , the best bound on the asymptotic complexity of a matrix multiplication algorithm is O( n 2.371552 ) .
The gradient of a function is obtained by raising the index of the differential , whose components are given by: =; =; =, = = The divergence of a vector field with components is
The sum here extends over all elements σ of the symmetric group S n, i.e. over all permutations of the numbers 1, 2, ..., n. This formula differs from the corresponding formula for the determinant only in that, in the determinant, each product is multiplied by the sign of the permutation σ while in this formula each product is unsigned. The ...
Its main application is to give an upper bound for the number of rational points of bounded height on or near algebraic varieties defined over the rational numbers. The main novelty of the determinant method is that in all incarnations, the estimates obtained are uniform with respect to the coefficients of the polynomials defining the variety ...