# Symmetric

<mathematics> 1. A relation R is symmetric if, for all x and y,

x R y

=>

y R x

If it is also antisymmetric (x R y & y R x => x == y) then x R y => x == y, i.e. no two different elements are related.

2. In linear algebra, a member of the tensor product of a vector space with itself one or more times, is symmetric if it is a fixed point of all of the linear isomorphisms of the tensor product generated by permutations of the ordering of the copies of the vector space as factors.

It is said to be antisymmetric precisely if the action of any of these linear maps, on the given tensor, is equivalent to multiplication by the sign of the permutation in question.