Definition
Let V be a vector space and
a tensor of order r. Then T is a symmetric tensor if
for the braiding maps associated to every permutation σ on the symbols {1,2,...,r} (or equivalently for every transposition on these symbols).
Given a basis {ei} of V, any symmetric tensor T of rank r can be written as
for some unique list of coefficients (the components of the tensor in the basis) that are symmetric on the indices. That is to say
for every permutation σ.
The space of all symmetric tensors of rank r defined on V is often denoted by Sr(V) or Symr(V). It is itself a vector space, and if V has dimension N then the dimension of Symr(V) is the binomial coefficient
Read more about this topic: Symmetric Tensor
Famous quotes containing the word definition:
“... we all know the wags definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on life (based on wording in the First Edition, 1935)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lensif we are unaware that women even have a historywe live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)