Is there a name for a matrix whose n-th power is the identity matrix

Question

I am new to this community so my apologies it this is a duplicate, feel free to flag it.

I am currently working on cyclically symmetric structural mechanics and we exploit the finite group linear representation.

In this theory, properties of rotation matrices are used, specifically the fact that for a rotation matrix $\mathbf{R}$ of angle $2\pi/N$, $\mathbf{R}^{N} = \mathbf{I}$.

Matrices such that $\mathbf{R}^{N} = \mathbf{0}$ are called nilpotent matrices, but is there a name for the mentioned rotation matrices ?

Answer 1

I would just call it a matrix of finite order.

Answer 2

I do not know a name for such matrices but ... we have $R^{N+1}=R$ then $R$ is a periodic matrix of period $N$. See here.

Then $R^N$ is idempotent, indeed $(R^N)^2=R^{2N}=R^{N+1}R^{N-1}=R^N$ and the only invertible idempotent matrix is identity. Then an equivalent notion is "invertible periodic matrix".