TITLE: Is there a clean way to write down this "product" of a vector and a matrix? Does it have any interpretation?
QUESTION [8 upvotes]: Let 
$$v = \begin{pmatrix}
  v_1 \\
  \vdots \\
  v_m
 \end{pmatrix}$$ and $$A = 
 \begin{pmatrix}
  a_{11} & \cdots & a_{1n} \\
  \vdots & \ddots &  \vdots\\
  a_{m1} & \cdots & a_{mn}
 \end{pmatrix}$$
If we define the operation $\Box$ as follows
$$v \Box A = \begin{pmatrix}
  v_1 a_{11} & \cdots & v_1 a_{1n} \\
  \vdots & \ddots &  \vdots\\
  v_m a_{m1} & \cdots & v_m a_{mn}
 \end{pmatrix},$$
is there a clean way to write it down? Does the operation have any interpretation?
I was hoping it has something to do with tensor products but, after looking up about tensors, it doesn't look like it.

REPLY [18 votes]: Let's define the diagonal matrix $V$ as follows:
$$V := \mbox{diag} (v) =
 \begin{pmatrix}
  v_{1} & 0 & \cdots & 0\\
 0 & v_{2} & \cdots & 0\\
  \vdots & \vdots & \ddots &  \vdots\\
  0 & 0 & \cdots & v_{m}
 \end{pmatrix}$$
Then:
$$v \Box A  = V \cdot A , $$
where $\cdot$ is the standard product between matrices.