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:
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)