Symbol for elementwise multiplication of vectors

(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.

• Wikipedia uses $\circ$ (\circ) to denote the Hadamard product (which is the operation you describe)
• This answer makes a good case for $\odot$ (\odot) being used instead.

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.