schiefsymmetrischer

Schiefsymmetrischer is a German adjective that translates to "skew-symmetric" or "anti-symmetric" in English. It is primarily used in mathematics, particularly in linear algebra and group theory, to describe matrices, tensors, and bilinear forms.

A matrix is considered schiefsymmetrischer if its transpose is equal to its negative. Mathematically, this means that for a matrix A, A^T = -A. This implies that the diagonal elements of a schiefsymmetrischer matrix must be zero. For example, the matrix [[0, 2], [-2, 0]] is schiefsymmetrischer because its transpose is [[0, -2], [2, 0]], which is equal to the negative of the original matrix.

In a similar fashion, a bilinear form B is schiefsymmetrischer if B(v, w) = -B(w, v) for all vectors v and w in the vector space. This property is closely related to the concept of skew-symmetric matrices, as the matrix representation of a schiefsymmetrischer bilinear form is a schiefsymmetrischer matrix.

The concept of schiefsymmetrischer also appears in abstract algebra, for instance, in the context of Lie algebras. A Lie algebra is a vector space equipped with a bilinear operation called the Lie bracket, denoted by [, ], which is schiefsymmetrischer and satisfies the Jacobi identity. The schiefsymmetrischer property of the Lie bracket means that [v, w] = -[w, v].

The term "schiefsymmetrischer" highlights the asymmetric nature of the relationship it describes, where swapping the order or position results in an inverse relationship.