What is the meaning of rotation of a matrix by complex exponential?

Question

I am reading the following theorem:

I dont understand what the $4^{th}$ and $5^{th}$ points are telling me? What is the meaning of the phrase "The spectrum of $A$ is invariant under rotation by $\exp{(i 2 \pi/P)}$"?

Answer 1

I think that the intended point is that the spectrum is invariant under multiplication by $\exp(i 2 \pi/p)$, which is to say that it is invariant under rotation by an angle of $2 \pi/p$.

In other words, if $\lambda \in \Bbb C$ is an eigenvalue of $A$, then so is $\exp(i 2 \pi /p)\lambda$.

I think that writing "rotation by $\exp(i 2\pi/p)$" was an error by the author.

Answer 2

4. says that if $\lambda$ is an eigen value then so is $e^{i2\pi /p} \lambda$. Since $1$ is an eigen value of any row-stochastic matrix $A$ it follows that when you multiply $1$ by any power of $e^{i2\pi /p}$ you get an eigen value of $A$.