What does antisymmetric mean?
Antisymmetric is a technical adjective from mathematics and physics describing objects whose behavior reverses sign when two components are exchanged. Applied to a binary relation, antisymmetry means that whenever a relates to b and b relates back to a, the two elements must be identical — the familiar 'less than or equal to' relation on numbers works this way. Applied to matrices, tensors, or functions of several variables, it means that swapping any two indices or arguments multiplies the quantity by −1, so entries above the diagonal are the negatives of those below. The term should not be confused with asymmetric, which merely denotes an absence of symmetry; antisymmetry is a precise, structured property rather than a simple lack of balance. It appears most often in linear algebra, order theory, and quantum mechanics, where the antisymmetry of fermionic wave functions underlies the Pauli exclusion principle.
adjectiveDescribing a relation, matrix, or function that changes sign when two arguments or indices are exchanged: formally, one for which swapping the elements reverses the sign (or negates the value). In mathematics and physics, an antisymmetric relation is one where if a is related to b and b is related to a, then a must equal b.
- Of a binary relation: one in which no two distinct elements are mutually related (if aRb and bRa, then a = b).
- Of a matrix or tensor: one whose entries change sign when two indices are swapped (Aᵢⱼ = −Aⱼᵢ).
- Of a function of several variables: one whose value negates when any two arguments are interchanged.
"The matrix was antisymmetric, so every entry above the diagonal was the negative of its mirror entry below."
"In set theory, the 'less than or equal to' relation on numbers is antisymmetric because x ≤ y and y ≤ x together imply x = y."
"Quantum mechanics requires the total wave function of identical fermions to be antisymmetric under particle exchange."
Swap two indices of an antisymmetric matrix and everything flips sign — mathematics' way of saying 'same parts, opposite attitude.'
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of antisymmetric
Antisymmetric combines the Greek-derived prefix anti-, meaning 'opposite' or 'against,' with symmetric, which traces back through Latin symmetricus to the Greek summetros ('of like measure'), built from syn- ('together') and metron ('measure'). The word arose as mathematicians formalized properties of relations and matrices in the nineteenth century, coining antisymmetric as the sign-reversing counterpart of symmetric. Its cognates include symmetric, asymmetric, and the broader family of measure-based terms descended from the Greek metron.
Related word forms
How antisymmetric is actually used
Strictly technical vocabulary used almost exclusively in mathematics, logic, physics, and computer science; rarely encountered in general prose. Note the subtlety that an antisymmetric relation may still hold between an element and itself — antisymmetry does not forbid reflexivity.
Easily confused with antisymmetric
Asymmetric simply means lacking symmetry, whereas antisymmetric describes a precise mathematical property where exchanging elements reverses the sign.
A symmetric object or relation remains unchanged under exchange of its parts, while an antisymmetric one changes sign or forces equality.