Symmetric tensor: Difference between revisions

Revision as of 12:30, 11 September 2013

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==Decomposition==

For symmetric tensors of arbitrary order ''k'', decompositions

:<math>T = \sum_{i=1}^r \lambda_i \, \underbracev_i^{v_i\otimes v_i\otimes\cdots\otimes v_i}_{k\text{ times}}</math>

are also possible. The minimum number ''r'' for which such a decomposition is possible is the ''symmetric'' [[Tensor_(intrinsic_definition)#Tensor_rank|rank]] of ''T''<ref name="Comon2008"></ref>. For second order tensors this corresponds to the rank of the matrix representing the tensor in any basis, and it is well-known that the maximum rank is equal to the dimension of the underlying vector space. However, for higher orders this need not hold: the rank can be higher than the number of dimensions in the underlying vector space. The [[higher-order singular value decomposition]] of a symmetric tensor is a special decomposition of this form <ref name="Comon2008">{{Cite doi|10.1137/060661569}}</ref> (often called the [[CP decomposition|canonical decomposition]].)