Suppose I have an $m \times n$ matrix $\mathbf{A}$, and a column $m$-vector $\mathbf{v}$. Let $\mathbf{B}$ represents the matrix resulted from the element-wise multiplication of each column of $\mathbf{A}$ with the vector $\mathbf{v}$. For example,
$$\mathbf{A} = \begin{bmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22} \\
a_{31} & a_{32} \\
\end{bmatrix}, \qquad
\mathbf{v} = \begin{bmatrix}
v_{1} \\
v_{2} \\
v_{3} \\
\end{bmatrix}, \qquad
\mathbf{B} = \begin{bmatrix}
a_{11} v_{1} & a_{12} v_{1} \\
a_{21} v_{2} & a_{22} v_{2} \\
a_{31} v_{3} & a_{32} v_{3}
\end{bmatrix}$$
What do we call this kind of multiplication? Is it a Hadamard product?
Best Answer
$$ {\bf B} = \operatorname{diag} ({\bf v}) \, {\bf A} $$