Compute the Matrix-Vector
Jelly, 5 bytes
ṙJṚæ.

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Explanation
Firstly:
$$\left[\begin{matrix}
\vec{v}_1 \\
\vec{v}_2 \\
\vec{v}_3 \\
\vec{v}_4 \\
\end{matrix}\right] \vec{x} =
\left[\begin{matrix}
\vec{v}_1 \cdot \vec{x} \\
\vec{v}_2 \cdot \vec{x} \\
\vec{v}_3 \cdot \vec{x} \\
\vec{v}_4 \cdot \vec{x} \\
\end{matrix}\right]$$
where \$\vec{v}_k\$ are row vectors and \$x\$ is a column vector.
This demonstrates that matrix multiplication is just dot product between rows and columns.
Then, \$\vec{v}_1\$ is actually \$\vec{v}\$ rotated \$0\$ to the right, and \$\vec{v}_k\$ is \$\vec{v}\$ rotated \$k-1\$ to the right, etc.
From another angle, \$\vec{v}_1\$ is \$\vec{v}\$ rotated \$n\$ to the left, and \$\vec{v}_n\$ is \$\vec{v}\$ rotated \$1\$ to the left, etc.
How it works
ṙJṚæ.   input: z (a list of length n)
ṙJ      [rot(z,1), rot(z,2), ..., rot(z,n)] (to the left)
  Ṛ     [rot(z,n), ..., rot(z,2), rot(z,1)]
   æ.   [rot(z,n).z , ..., rot(z,2).z , rot(z,1).z] (dot product)