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.

(i) The identity relation on a set A is an antisymmetric relation.
(ii) Let R be a relation on the set N of natural numbers defined by
x R y ⇔ 'x divides y' for all x, y ∈ N.
This relation is an antisymmetric relation on N.

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.

Antisymmetric

Meaning

Adj

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