Does this type of "cyclic" matrix have a name? Let $\{a_1, a_2, ..., a_n\} \subset \mathbb{C}$ and consider the matrix of the form
$$
\begin{bmatrix}
  a_1 & a_2 & ... & a_{n-1} & a_n\\
  a_2 & a_3 & ... & a_n & a_1\\
  .\\
  .\\
  .\\
  a_n & a_1 & ... & a_{n-2} & a_{n-1}
\end{bmatrix}
$$
Does this type of matrix have a specific name?

Yes, it is a type of Circulant Matrix. 
