1
$\begingroup$

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?

Improve this question
$\endgroup$
4
  • 2
    $\begingroup$ It's not unlikely that this comes up in tropical geometry (en.wikipedia.org/wiki/Tropical_geometry), where the "product" is the original addition. $\endgroup$
    – Tom De Medts
    Commented Jun 17, 2013 at 14:42
  • $\begingroup$ "Outer sum"? I think that's what the programming language APL called it. $\endgroup$
    – Deane Yang
    Commented Jun 17, 2013 at 15:53
  • $\begingroup$ I've occasionally heard "tensor sum". $\endgroup$
    – Noah Stein
    Commented Jun 17, 2013 at 17:28
  • $\begingroup$ Thank you Tom for direction. According to personalpages.manchester.ac.uk/staff/Marianne.Johnson/… It is indeed tropical product of two vectors $a \otimes b^T$. $\endgroup$
    – Fedor Nikitin
    Commented Jun 17, 2013 at 17:52

2 Answers 2

Reset to default
1
$\begingroup$

It is tropical product of vectors $x$ and $y^T$, i.e. $x \otimes y^T$.

See http://personalpages.manchester.ac.uk/staff/Marianne.Johnson/Peter_Butkovic_Beamer%20Manchester%202012%20vs2%20print.pdf

Improve this answer
$\endgroup$
0
$\begingroup$

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.

Improve this answer
$\endgroup$
3
  • $\begingroup$ There are two vectors of ones $e_m$ and $e_n$ should be above because I assume that $x$ and $y$ are of dimensions. $\endgroup$
    – Fedor Nikitin
    Commented Jun 17, 2013 at 20:11
  • $\begingroup$ Yes, you are correct. $\endgroup$
    – Federico Poloni
    Commented Jun 18, 2013 at 6:24
  • $\begingroup$ Just wanted to add, that this operation appears naturally in the context of optimal transport (popped up in doi.org/10.1007/s00245-019-09614-w e.g.). $\endgroup$
    – Dirk
    Commented Jun 23, 2023 at 9:14

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .