symmetric
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.
(1996-09-22)