Search results
Results from the WOW.Com Content Network
In mathematics, the Bareiss algorithm, named after Erwin Bareiss, is an algorithm to calculate the determinant or the echelon form of a matrix with integer entries using only integer arithmetic; any divisions that are performed are guaranteed to be exact (there is no remainder).
In mathematics, Dodgson condensation or method of contractants is a method of computing the determinants of square matrices.It is named for its inventor, Charles Lutwidge Dodgson (better known by his pseudonym, as Lewis Carroll, the popular author), who discovered it in 1866. [1]
Instead, the determinant can be evaluated in () operations by forming the LU decomposition = (typically via Gaussian elimination or similar methods), in which case = and the determinants of the triangular matrices and are simply the products of their diagonal entries. (In practical applications of numerical linear algebra, however, explicit ...
Laplace (1772) gave the general method of expanding a determinant in terms of its complementary minors: Vandermonde had already given a special case. [29] Immediately following, Lagrange (1773) treated determinants of the second and third order and applied it to questions of elimination theory; he proved many special cases of general identities.
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.
In two dimensions, the Levi-Civita symbol is defined by: = {+ (,) = (,) (,) = (,) = The values can be arranged into a 2 × 2 antisymmetric matrix: = (). Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like supersymmetry [1] and twistor theory, [2] where it appears in the context of 2-spinors.
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
where | g | is the absolute value of the determinant of the matrix of scalar coefficients of the metric tensor . These are useful when dealing with divergences and Laplacians (see below). The covariant derivative of a vector field with components is given by: