antisymmetric
形容词 adj.
英文释义
形容词 adj.
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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.
— 1987, David C. Buchthal, Douglas E. Cameron, Modern Abstract Algebra, Prindle, Weber & Schmidt, page 479, The standard example for an antisymmetric relation is the relation less than or equal to on the real number system.
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Whose sign changes on the application of a matrix transpose or some generalisation thereof:; 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, #43;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.
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Whose sign changes on the application of a matrix transpose or some generalisation thereof:; That changes sign when any two indices are interchanged (e.g., Tᵢⱼₖ = -Tⱼᵢₖ);
— Notice that the tensors defined by: #92;textstyleT#95;S#92;equiv#92;frac#123;1#125;#123;2#125;(T#43;Tᵀ), #92;textstyleT#95;A#92;equiv#92;frac#123;1#125;#123;2#125;(T-Tᵀ), (3.47) are the symmetric and antisymmetric parts, respectively; they are known as the symmetric and antisymmetric parts of T.
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Whose sign changes on the application of a matrix transpose or some generalisation thereof:; For which B(w,v) = -B(v,w).
— Antisymmetric bilinear forms and wedge products are defined exactly as above, only now they are functions from #92;Rⁿ#92;times#92;Rⁿ to #92;R.[…] Exercise 21 Show that every antisymmetric bilinear form on #92;R³ is a wedge product of two covectors.
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Etymology tree
Proto-Indo-European *h₂ent-
Proto-Indo-European *-s
Proto-Indo-European *h₂énts
Proto-Indo-European *-i
Proto-Indo-European *h₂énti
Ancient Greek ᾰ̓ντῐ́ (ăntĭ́)
Ancient Greek ἀντι- (anti-)der.
English anti-
English symmetric
English antisymmetric
From anti- + symmetric.
Proto-Indo-European *h₂ent-
Proto-Indo-European *-s
Proto-Indo-European *h₂énts
Proto-Indo-European *-i
Proto-Indo-European *h₂énti
Ancient Greek ᾰ̓ντῐ́ (ăntĭ́)
Ancient Greek ἀντι- (anti-)der.
English anti-
English symmetric
English antisymmetric
From anti- + symmetric.
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