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I am wondering if there is any standard notation for the element-wise division of vectors.

I am going to use $\oslash$ for this purpose, similar to $\odot$ that is used in some texts for element-wise multiplication. For example, assuming $\vec{u}$ and $\vec{v}$ are vectors of length $k$, then $\vec{x} = \vec{u} \oslash \vec{v}$ if $\vec{x}(i) = \vec{u}(i) / \vec{v}(i)$. Would that be strange to use this in a scientific paper?

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    $\begingroup$ Typically, I would shy away from introducing new notation. You can accomplish the same result by left-multiplying the vector by a diagonal matrix whose entries are the divisors. $\endgroup$
    – Emily
    Jul 18, 2012 at 4:30
  • $\begingroup$ Octave/Matlab use $u./v$. Not pretty, but might be more recognizable, depending on your audience. $\endgroup$
    – copper.hat
    Jul 18, 2012 at 4:32
  • $\begingroup$ I agree with @Ed. If you feel like dividing or multiplying vectors componentwise, they are not vectors to you any more. They are diagonal matrices. Notice that you don't lose anything by switching to matrices: addition, scalar multiplication and scalar product still work fine. You can write diag(a,b,c) when the entries need to be listed. $\endgroup$
    – user31373
    Jul 18, 2012 at 5:00

3 Answers 3

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That is known as The Hadamard division and is documented here:

https://en.wikipedia.org/wiki/Hadamard_product_(matrices)#Analogous_operations

as:

$\begin{align} C &= A {\oslash} B \\ C_{ij} &= A_{ij} / B_{ij} \end{align}$

There are some other useful Hadamard operations and notations listed there.

It is noteworthy that the Hadamard multiplication uses the symbol "$\circ$" not "$\otimes$" or "$\odot$".

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  • $\begingroup$ Given this is still being found and liked, I should note that since 2018, Wikipedia has changed. On 22 August 2019, 69.128.108.106 added the ⊙ notation as acceptable as well, and on 25 October 2023,Limelike Curves removed the ∘ notation from the definition. So it seems ⊙ is now the accepted symbol for the Hadamard multiplication. Inasmuch as Wikipedia reflects trends in notation accurately. $\endgroup$ Apr 9 at 4:59
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I have no idea about standard notations regarding this operation, but I would recommend expressing $$(x_1,\ldots,x_n)\oslash(y_1,\ldots,y_n)$$ instead as $$(x_1,\ldots,x_n)\odot(\tfrac{1}{y_1},\ldots,\tfrac{1}{y_n})$$ After all, the notation $\oslash$ will not make sense unless every $y_i\neq0$.

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    $\begingroup$ The only problem I see with this recommendation is that it is desirable at times to express the operation without reference to the vectors elements, namely like: $\mathbf x \oslash \mathbf y$ and this formulation would appear to have no added value: $\mathbf x \odot \frac {1} {\mathbf y}$. $\endgroup$ Jun 20, 2018 at 0:34
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Maybe you can try to take advantage of a diagonal matrix for expression: $\text{diag}^{-1}(\vec{v})\,\vec{u}$, where $\text{diag}(\cdot)$ returns a matrix with the vector as its diagonal and "$^{-1}$" means inverse of the matrix.

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