Antisymmetry

Redirect to:

• Multilinear algebra

The opposite of the symmetry property of functions of several variables. Instead of preserving, flipping variables flips the sign of the function.

Suppose $X$ is a set (possibly a vector space) and suppose we have a function: $$\varphi \colon X^k \rightarrow {\bf R},$$ $$\varphi (x,y,z,...)= r \in {\bf R}.$$

It is called anti-symmetric if

• $\varphi(x,y,z,...) = -\varphi(y,x,z,...),$
• $\varphi(x,y,z,...) = -\varphi(x,z,y,...),$
• $\varphi(x,y,z,...) = -\varphi(z,y,x,...).$
• etc.

As an example, see differential forms.