How to express in linear algebra notation the operation of scaling a matrix by each element of a vector?

Question

Define vector ${\mathbf{a}} = \left( {\begin{array}{*{20}{c}} a_{1} \\ a_{2} \\ \vdots\\ a_{n} \end{array}} \right)$, and matrix ${\mathbf{B}} = \left( {\begin{array}{*{20}{c c c c}} b_{11} & b_{12} & \dots & b_{1m}\\ b_{21} & b_{22} & \dots & b_{2m}\\ \vdots & \vdots & \ddots & \vdots \\ b_{n1} & b_{n2} & \dots & \ b_{nm} \end{array}} \right)$, I would like to express in linear algebra notation, ${\mathbf{B}}$ scaled by each element of ${\mathbf{a}}$, i.e, $\left( {\begin{array}{*{20}{c c c c}} a_{1}*b_{11} & a_{1}*b_{12} & \dots & a_{1}*b_{1m}\\ a_{2}*b_{21} & a_{2}*b_{22} & \dots & a_{2}*b_{2m}\\ \vdots & \vdots & \ddots & \vdots \\ a_{n}*b_{n1} & a_{n}*b_{n2} & \dots & \ a_{n}*b_{nm} \end{array}} \right)$.

Answer 1

$(B^tA_n)^t = A_nB$ where $A_n$ is a diagonal matrix having diagonal $a_1, a_2,...,a_n$.