English Dictionary

antisymmetric

形容詞

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(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.

日本語の意味

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。

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

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。

意味（1）

(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.

The standard example for an antisymmetric relation is the relation less than or equal to on the real number system.

意味（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.

意味（3）

(of a tensor) That changes sign when any two indices are interchanged (e.g., Tᵢⱼₖ = -Tⱼᵢₖ);

T_S≡1/2(T+Tᵀ), T_A≡1/2(T-Tᵀ), (3.47) are the symmetric and antisymmetric parts, respectively; they are known as the symmetric and antisymmetric parts of T.

意味（4）

(of a bilinear form) For which B(w,v) = -B(v,w).

Exercise 21 Show that every antisymmetric bilinear form on R ³ is a wedge product of two covectors.