# what are some properties of the diag operator?

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.)

-
 Oh, so you want the Hadamard product of two vectors? – J. M. Sep 12 '11 at 15:02 I would name it "pointwise multiplication". $\mathbb R^n$ under pointwise addition and multiplication forms a ring. We can identify the vectors $(c,c,\ldots,c)$ for any $c\in \mathbb R$ with $\mathbb R$ itself; with this identification, the multiplication between scalars and vectors agree with the ordinary vector space structure of $\mathbb R^n$. – Henning Makholm Sep 12 '11 at 15:06 ah ha... yes, that's exactly it. any idea about my question w.r.t. nesting and distributivity? – distbeta Sep 12 '11 at 15:08 I thought about it some more. It seems to be hopeless to get any kind of properties – distbeta Sep 12 '11 at 16:38