How to represent Hadamard product in terms of matrix multiplication? In the case of two vectors $u, v$ with dimensions $n\times 1$, their Hadamard product can be represented by the following matrix multiplication:
$$\mathrm{diag}(u)v = \left[\begin{array}{ccc}
u_{1} &amp;  &amp; 0\\
 &amp; \ddots\\
0 &amp;  &amp; u_{n}
\end{array}\right]\left[\begin{array}{c}
v_{1}\\
\vdots\\
v_{n}
\end{array}\right]=\left[\begin{array}{c}
u_{1}v_{1}\\
\vdots\\
u_{n}v_{n}
\end{array}\right]\equiv u\circ v$$

Is there a way to generalize this for Hadamard products of matrices?
Let $\mathcal M_n$ be the space of $n\times n$ matrices (over some field). Then if $A \in \mathcal M_n$ you can define a linear map:
$ T_A : \mathcal M_n \to \mathcal M_n$ given by $T_A(B) = A\circ B$, where $\circ$ is the Hadamard/Schur product of $A$ and $B$ (note that this definition is basis dependent!). The matrix units are eigenvectors of this linear trasformation:
$$ T_A(E_{ij}) = a_{ij}E_{ij},$$
where $A=(a_{ij})_{i,j=1\dots n}$.

Now $T_A$ is a linear operator, so if you pick a basis for $\mathcal M_n$ you get a matrix representation for it. If you choose the same basis in which you defined the Hadamard product, you get a diagonal matrix of dimension $n^2\times n^2$ whose diagonal entries are the matrix entrires of $A$, i.e. the $a_{ij}$ (ordered according of the order of the matrix units you choose).
Hadamard product (Schur product) of matrices is element-wise product (two matrices dimension have to be same). 

Reference: Topics in matrix analysis.