Simple question. Can one write every second rank tensor $T^{ab}$ as some finite sum $\sum U^aV^b$ with $U^a$, $V^b$ tensors? Apologies if this is an incredibly standard result - I don't own a textbook on tensors! If this indeed is true, then presumably it generalises to tensors of all ranks?
Relationship between Tensors of Different Rank
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1$U^a \otimes V^b$ isn't a first-rank tensor (depending on what exactly you mean by "rank"). – 2012-05-22
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0You are probably after this notion: http://en.wikipedia.org/wiki/Tensor_product#Tensor_product_of_two_tensors – 2012-05-22
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3There are an obvious set of $nm$ basis vectors for the space of $n\times m$ matrices, each given by $1$ in the $(i,j)$ entry and zero everywhere else, and each of these can be given the expression $e_i e_j^T$, where $e_i,e_j$ are basis column vectors of $n$ and $m$ dim vector spaces. Is this what you are after? – 2012-05-22
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0@QiaochuYuan: Sorry the question was unclear - I've reworded it now – 2012-05-22
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0@GiuseppeNegro: Yeah looking at that I guess the question is can I write every 2 tensor as the tensor product of two 1 tensors – 2012-05-22
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0@anon: That seems to prove what I was after - thanks! – 2012-05-22