Let $v_1,...,v_n$ and $u_1,...,u_n$ be two sequences of $n$ numbers, and define
$u \cdot v = \operatorname{diag}(u)\cdot\operatorname{diag}(v)\cdot 1^t$
i.e. --
$u \cdot v = w$ where $w$ is a vector of length $n$ such that $w_i = u_i \cdot v_i$
(a vector which each coordinate is a multiplication of the two corresponding coordinates in $u$ and $v$).
I would like to know some properties of this operator. I have an expression which uses this operator, and I would like to try and simplify it somehow.
I couldn't find many properties of $\operatorname{diag}$ in wikipedia -- knowing such properties could also shed some light on this.
I am especially interested in properties of nesting and distribiutivity with respect to matrix multiplication, i.e.
$(u A \cdot v B) \cdot w = ?$
where $A$ and $B$ are matrices.
(knowing also whether this $\cdot$ operator has a well-known name would also help.)
