Is there a formal name for this matrix? I've been using a matrix of the following form:
$$
\begin{bmatrix}
-1 & 1 & 0 \\
0 & -1 & 1 \\
1 & 0 & -1
\end{bmatrix}
$$
Which is just a circularly-shifted $I$ matrix minus another $I$ matrix. Essentially, a permutation matrix minus an identity. Is there a formal name for that?

Not that I have heard. It is a circulant matrix, though, and the permutation matrix alone without the $-I$ is sometimes called the cyclic shift matrix, circulant generator or generator of the circulant algebra.
