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Suppose we have two vectors $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^m$. I could define the mapping $$ T: \mathbb{R}^n\times \mathbb{R}^m \rightarrow \mathbb{R}^{n\times m} $$ as follows $$ T(x,y) = ( x_i+y_j )_{i,j=1}^{n,m}, $$ i.e. $T(x,y)$ is $n\times m$ matrix with elements equal sum of corresponding vectors. If instead of $x_i+y_j$ one considers $x_i \cdot y_j$ then it is matrix multiplication of two vectors ($x\cdot y^T$, where $x,y$ are column vectors). But what about sum?

Is that operation known and studied somewhere?

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It's not unlikely that this comes up in tropical geometry (, where the "product" is the original addition. – Tom De Medts Jun 17 '13 at 14:42
"Outer sum"? I think that's what the programming language APL called it. – Deane Yang Jun 17 '13 at 15:53
I've occasionally heard "tensor sum". – Noah Stein Jun 17 '13 at 17:28
Thank you Tom for direction. According to… It is indeed tropical product of two vectors $a \otimes b^T$. – Fedor Nikitin Jun 17 '13 at 17:52

2 Answers 2

I'd write your operation simply as $x e^T + ey^T$, where $e$ is the vector of all ones. I have never encountered this operation directly studied in a research context (my guess is that it's simply "not interesting enough", since it's a small variation to the common theme of a rank-2 matrix), but I have seen several times the related matrix with elements $$ T_{ij}= \frac{1}{x_i - y_j}, $$ which is known as Cauchy matrix.

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There are two vectors of ones $e_m$ and $e_n$ should be above because I assume that $x$ and $y$ are of dimensions. – Fedor Nikitin Jun 17 '13 at 20:11
Yes, you are correct. – Federico Poloni Jun 18 '13 at 6:24

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