Symmetric Tensor - Definition

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:

    It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.
    François, Duc De La Rochefoucauld (1613–1680)

    Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.
    Nadine Gordimer (b. 1923)

    ... 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 lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.
    Adrienne Rich (b. 1929)