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This is a notation question. Assume one is given two vector $\mathbf{a}$ and $\mathbf{b}$, and one constructs a third vector $\mathbf{c}$ whose elements are given by $$c_k=a_k b_k$$ Is there any standard notation for this simple operation? Is the notation below acceptable? $$\mathbf{c}=\mathbf{a}\otimes \mathbf{b}$$

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3 Answers 3

up vote 12 down vote accepted

(Minor edits.)

It turns out that the symbol $\odot$ is often used to denote component-wise multiplication (a few examples are given in the comments below); $\circ$ and $*$ are common alternatives.

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Example 1: Last sentence in page 29 here: columbia.edu/~nsa2106/Aybat_Paper4_FALC.pdf –  Shai Covo Jul 20 '11 at 6:33
    
Example 2: The sentence above Equation (8) here: uow.edu.au/~mwand/publicns/Gangu07.pdf –  Shai Covo Jul 20 '11 at 6:38
    
Example 3: Line 10 on page 14 here stanford.edu/~tsachy/pdf_files/…. –  Shai Covo Jul 20 '11 at 6:43
    
Example 4: Last sentence on page 16 here: alphard.ethz.ch/Hafner/Workshop/Sandfort2010.pdf –  Shai Covo Jul 20 '11 at 6:47
    
Example 5: End of page 2 here: tu-ilmenau.de/fakia/fileadmin/template/startIA/neuroinformatik/… –  Shai Covo Jul 20 '11 at 6:50

No, I would be concerned about $\otimes$ causing confusion with the outer product (although the outer product will produce a matrix, and the componentwise product will produce a vector, so if the context is clear enough perhaps this will not be a problem).

I recommend writing componentwise multiplication of vectors using some symbol that does not have a standard meaning, perhaps $\star$ (\star) or $\diamond$ (\diamond), so that people reading won't have any preconceptions about what might be meant.

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Additionally, $\otimes$ is also often used for the Kronecker product, so using that to denote the Hadamard product would be quite the symbol overload... –  J. M. Jul 20 '11 at 11:16

If I ever needed to perform a Hadamard product of two vectors $\mathbf a$ and $\mathbf b$, apart from the usual MATLAB notation (as mentioned in the first linked question in the comments), I'd probably use $\mathrm{diag}(\mathbf a)\cdot\mathbf b$, where $\mathrm{diag}(\mathbf a)$ is the diagonal matrix with diagonal entries $a_k$.

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