antisymmetric

Adjective

not-comparable of a binary relation R on a set S of a matrix of certain mathematical objects of a tensor of a bilinear form

(set theory, order theory, of a binary relation R on a set S) Having the property that, for any two distinct elements of S, at least one is not related to the other via R; equivalently, having the property that, for any x, y ∈ S, if both xRy and yRx then x=y.

Japanese Meaning

反対称的：集合S上の2項関係Rが、任意の異なる2要素について、少なくとも一方が他方と関係していない、もしくは逆に、どちらか一方が他方と関係しているときは、必ず両者が同一である（すなわち、xRyかつyRxならばx=yとなる）という性質を持つこと。

What is this buttons?

In the poset example, the relation R is antisymmetric, so if aRb and bRa hold then a and b must be equal.

Google Translation / DeepL Translation

順序集合の例では、関係Rは反対称であり、集合の異なる任意の2要素aとbについて少なくとも一方がもう一方と関係していないという性質を持つため、もしaRbとbRaが両方成り立つならaとbは等しい。

Details

Sense（1）

Details

Sense（2）

(of a matrix) Whose transpose equals its negative (i.e., Mᵀ = −M);

The eigenvalues of an antisymmetric matrix are all purely imaginary numbers, and occur as conjugate pairs, +iw and -iw. As a corollary it follows that an antisymmetric matrix of odd order necessarily has one eigenvalue equal to zero; antisymmetric matrices of odd order are singular.

Google Translation / DeepL Translation