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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?

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$U^a \otimes V^b$ isn't a first-rank tensor (depending on what exactly you mean by "rank"). –  Qiaochu Yuan May 22 '12 at 16:18
    
You are probably after this notion: en.wikipedia.org/wiki/… –  Giuseppe Negro May 22 '12 at 16:19
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There 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? –  anon May 22 '12 at 16:19
    
@QiaochuYuan: Sorry the question was unclear - I've reworded it now –  Edward Hughes May 22 '12 at 16:25
    
@GiuseppeNegro: Yeah looking at that I guess the question is can I write every 2 tensor as the tensor product of two 1 tensors –  Edward Hughes May 22 '12 at 16:25
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