Talk:Hadamard product (matrices)

Symbol notation
This post makes a case for using $$\odot$$ (instead of e.g. $$\circ$$) for the Hadamard product. The main arguments of the post are that it is not an overloaded symbol and that the dot captures the aspect of multiplication. The symbol $$\odot$$ is also frequently used in literature (as are $$\circ$$ and $$*$$). Perhaps the notation in this article could adopt the $$\odot$$ symbol? 84.27.170.190 (talk) 14:10, 17 May 2018 (UTC)
 * After giving this some thought, I think mentioning that $$\odot$$ is an alternative notation might be a good option. For example adding a sentence to the definition. I have also seen papers using $$\odot$$ instead of $$\circ$$ and at my university $$\odot$$ is used (at least) in undergrad maths lectures. On the other hand, imho, the readability of this particular article is much better using $$\circ$$ as compared to $$\odot$$ due to it's size. Also the referenced articles/books, or at least some of them (didn't check all) use $$\circ$$. In addition I don't know any linear algebra book using $$\odot$$ in their definition (which certainly doesn't mean there aren't any). Maybe someone with more experience/a more comprehensive maths background can comment on this? Ception (talk) 14:29, 10 July 2018 (UTC)

I totally agree with the first post that $$\odot$$ is much better suited and the votes on stack exchange support that strongly. The comment about the element-wise division makes a very good point as well. I think $$\odot$$ should be the standard notation of this article. --NeinKob (talk) 22:30, 20 February 2020 (UTC)

Element-wise matrix multiplication
A google search for "element-wise matrix multiplication" rightly leads to this article, but this nomenclature is never used in the article. Perhaps it could be added in the list of alternative names, since "entrywise product" is also mentioned? 84.27.170.190 (talk) 14:10, 17 May 2018 (UTC)
 * I think element-wise and entrywise is essentially the same. A user reading entrywise should be expected to recognize that he/she found what they were looking for. And since there is a citation for entrywise, I'd suggest leaving it that way. Ception (talk) 14:29, 10 July 2018 (UTC)
 * I couldn't find the term "element-wise" or “element wise” in the cited book https://doi.org/10.1017/CBO9780511810817
 * I think this term should be removed.
 * Also, the Wikipedia page says it's in ch5 that we find “entrywise product”/“Schur product” but it's actually in chapter 7.5
 * Should I edit myself? Nicolas22031998 (talk) 09:34, 20 June 2024 (UTC)

Repeated product symbol cause for confusion in this context: ordinary product or Hadamard product?
In the article I read:

"If A and B are positive-definite matrices, then the following inequality involving the Hadamard product is valid:[8]

$$\prod_{i=k}^n \lambda_i(\mathbf{A} \circ \mathbf{B}) \ge \prod_{i=k}^n \lambda_i(\mathbf{A} \mathbf{B}),\quad k=1,\ldots,n,$$

where $λ_{i}(A)$ is the $i$th largest eigenvalue of $A$."

In the context of this discussion of the Hadamard product, the notation leaves me wondering about its meaning in two ways; it looks ambiguous to me.

$$(\prod_{i=k}^n \lambda_i)(\mathbf{A} \circ \mathbf{B}) \ge (\prod_{i=k}^n \lambda_i)(\mathbf{A} \mathbf{B}),\quad k=1,\ldots,n,$$
 * Does the repeated multiplication just include the eigenvalues, or also the matrices? I.e., is this meant:

or is this meant:

$$\prod_{i=k}^n (\lambda_i(\mathbf{A} \circ \mathbf{B})) \ge \prod_{i=k}^n (\lambda_i(\mathbf{A} \mathbf{B})),\quad k=1,\ldots,n$$?
 * If the latter part of the previous question should be answered affirmatively, does the repeated product symbol here imply the usual matrix product or the Hadamard product? I.e., is this meant:

$${}^{\odot}\prod_{i=k}^n (\lambda_i(\mathbf{A} \circ \mathbf{B})) \ge {}^{\odot}\prod_{i=k}^n (\lambda_i(\mathbf{A} \mathbf{B})),\quad k=1,\ldots,n,$$?

I know not of a standard notation for repeated Hadamard products, so please excuse the extravagant notation.Redav (talk) 09:27, 16 May 2020 (UTC)


 * Ерунда какая то 213.151.30.2 (talk) 20:05, 15 April 2024 (UTC)

Bilinear map
It could be added that the operation is a bilinear map. 1234qwer1234qwer4 (talk) 15:50, 2 August 2020 (UTC)

Only matrices, or also vectors/tensors?
The article (both title and content) suggests that the name "Hadamard product" is only used for matrices. However, a similar element-wise/entrywise product could be applied to vectors, or to multilinear tensors (multidimensional arrays), not just to the specific case of matrices (two-dimensional arrays). Would those operations also be referred to as "Hadamard product", or is "Hadamard product" only used for the particular case of matrices?

If it's the former, the article should be rewritten, removing the implication that this is exclusive to matrices (or clarifying that "matrix" is being used as a generalized tensor). If it's the latter, then the article shouldn't state that this is synonymous with element-wise and entrywise product (or at least clarify that it's the entrywise product of matrices), and Entrywise product should not redirect here; also, if an article describing the generalized element-wise operation for any dimension exists, it should be linked. —Cousteau (talk) 08:30, 8 May 2024 (UTC)