Is there an existing name for "piecewise vector multiplication" 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
$$u ? v = [v_1 \times u_1, v_2 \times u_2, v_3\times u_3, ..., v_n \times u_n ]$$
For want of a better name, I have termed it "piecewise vector multiplication".
What is this operation normally known as in the literature?
 A: It's pointwise product. See Wikipedia articles here and here
A: 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.
A: That is the Hadamard product---which usually, though, is only used with matrices of a more matrixy shape.
A: 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.
