Given two vectors of size n u = [u_1, u_2, u_3, ..., u_n ] and v = [v_1, v_2, v_3, ..., v_n ]

What is the name of the operation "u ? v" such that the result is a vector of size n of the form u ? v = [v_1.u_1, v_2.u_2, v_3.u_3, ..., v_n.u_n ]

For want of a better name, I have termed it "piecewise vector multiplication".

What is this operation normally known as in the literature?

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    $\begingroup$ "Componentwise" or "coordinatewise" sounds more like standard terminology than "piecewise", but I don't know a really common name for this. $\endgroup$ – darij grinberg Dec 17 '09 at 11:40

It's pointwise product. See Wikipedia articles here and here

  • $\begingroup$ I have chosen this as the correct answer since the links provided better information than what is produced by a search for the (more) correct answer "Hadamard product". For example, the Mathworld article on the Hadamard product is just confusing in this context. $\endgroup$ – Niall Murphy Dec 17 '09 at 17:55

That is the Hadamard product---which usually, though, is only used with matrices of a more matrixy shape.

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    $\begingroup$ I heard a seminar by a functional analyst where this product figured quite prominently, and throughout which it was referred to systematically as «the wrong product»... $\endgroup$ – Mariano Suárez-Álvarez Dec 17 '09 at 12:00
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    $\begingroup$ "The freshman product". :-) $\endgroup$ – Greg Kuperberg Dec 17 '09 at 14:35

If $k$ is a field, the vector space $k^n$ endowed with the componentwise multiplication is called a diagonal algebra ( and so are isomorphic algebras). The terminology is due to Bourbaki and is justified by the following result.

If $M$ is a square matrix over $k$, it is diagonalizable over $k$ if and only if the algebra $k[M]$ is diagonal.The proof results from the diagonalization criterion (the minimal polynomial of $M$ should be split over $k$ and have distinct roots) , the isomorphism of $k$-algebras $ \frac {k[X]} {polmin_M (X)}\to k[M]$ and the Chinese remainder theorem.

These algebras are important to algebraic geometers because they are a model for étale algebras over a field. Indeed, a $k$ -algebra $A$ is étale if and only it becomes diagonal after some extension of the base field . More explicitly $A$ is étale if and only if for some field extension $K/k$ we have an isomorphism $A\otimes_k K \simeq K^n$ of K-algebras.

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    $\begingroup$ +1 mainly for the Bourbaki mention (also because this is correct.) $\endgroup$ – Harry Gindi Dec 17 '09 at 15:07

If you look at u and v as functions on the set S={1,2,3,...,n}, then they are elements of a function space on S, which is an algebra under pointwise addition and multiplication.


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