Symmetric tensor: Difference between revisions

Revision as of 01:22, 27 February 2016

Revision as of 01:23, 27 February 2016

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Given a [[basis (linear algebra)|basis]] {''e''^''i''} of ''V'', any symmetric tensor ''T'' of rank ''k'' can be written as

:<math>T = \sum_{i_1,\dotsldots,i_k=1}^N T_{i_1i_2\dotscdots i_k} e^{i_1} \otimes e^{i_2}\otimes\cdots \otimes e^{i_k}</math>

for some unique list of coefficients <math>T_{i_1i_2\dotscdots i_k}</math> (the ''components'' of the tensor in the basis) that are symmetric on the indices. That is to say

:<math>T_{i_{\sigma 1}i_{\sigma 2}\dotscdots i_{\sigma k}} = T_{i_1i_2\dotscdots i_k}</math>

for every [[permutation]] &sigma;.

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The space of all symmetric tensors of order ''k'' defined on ''V'' is often denoted by ''S''^''k''(''V'') or Sym^''k''(''V''). It is itself a vector space, and if ''V'' has dimension ''N'' then the dimension of Sym^''k''(''V'') is the [[binomial coefficient]]

:<math>\dim\, \operatorname{Sym}^k(V) = {N + k - 1 \choose k}.</math>

We then construct Sym(''V'') as the [[direct sum of vector spaces|direct sum]] of Sym^''k''(''V'') for ''k'' = 0,1,2,…