Antisimétrica

Antisimétrica is a term used in mathematics, particularly in linear algebra and abstract algebra, to describe certain types of functions or matrices. A function f is antisymmetric if swapping the arguments negates the result, meaning f(x, y) = -f(y, x) for all x and y in its domain. A common example is the cross product of two vectors in three-dimensional Euclidean space, where a × b = -(b × a).

In the context of matrices, a square matrix A is called antisymmetric if its transpose is equal to its negative. Mathematically, this is expressed as Aᵀ = -A. This implies that the diagonal elements of an antisymmetric matrix must be zero, because for any diagonal element aᵢᵢ, we have aᵢᵢ = -aᵢᵢ, which only holds true if aᵢᵢ = 0. The off-diagonal elements satisfy the relationship aᵢⱼ = -aⱼᵢ. For instance, the matrix [[0, 2], [-2, 0]] is antisymmetric. Antisymmetric matrices have important properties in various areas of mathematics and physics, including the study of rotations and the representation of Lie algebras.