English Dictionary

antisymmetric

adj

(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 relation R is antisymmetric if for all a and b in the set A, if (a, b) ∈ R and (b, a) ∈ R, then a = b.

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)

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.