Complex numbers/Multiplication/Real matrix/Exercise

Let ${}z\in \mathbb {C}$ be a complex number and let

$\mathbb {C} \longrightarrow \mathbb {C} ,w\longmapsto zw,$

be the multiplication map, which is a ${}\mathbb {C}$-linear map. How does the matrix related to this map with respect to the real basis ${}1$ and ${}{\mathrm {i} }$ look like? Let ${}z_{1}$ and ${}z_{2}$ be complex numbers with corresponding real matrices ${}M_{1}$ and ${}M_{2}$. Prove that the matrix product ${}M_{2}\circ M_{1}$ is the real matrix corresponding to ${}z_{1}z_{2}$.

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