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For symmetric tensors of arbitrary order ''k'', decompositions
:<math>T = \sum_{i=1}^r \lambda_i \, v_i^{\otimes k}</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">{{Cite doi|10.1137/060661569}}</ref> This minimal decomposition is called a Waring decomposition; it is a symmetric form of the [[tensor rank decomposition]]. 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.
==See also==
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