Eigenspaces of $s: v_1 \otimes v_2 \mapsto v_2 \otimes v_1 $

Question

Consider the linear „swap“ operator $s: v_1 \otimes v_2 \mapsto v_2 \otimes v_1 $, for $v_1 \otimes v_2 \in \mathbb{C}^n \otimes \mathbb{C}^n$
I am supposed to show that its eigenspaces are invariant subspaces of $\mathbb{C}^n \otimes \mathbb{C}^n$ under some other linear operator. In the solutions to my exercise they only consider the cases $s v_{\pm}=\pm v_{\pm}$ for $v_{\pm} \in \mathbb{C}^n \otimes \mathbb{C}^n $, so it seems as if the only possible eigenvalues of s are $\pm 1$, could someone maybe explain why that is the case?

Answer 1

Note that $s$ is its own inverse. Let $v$ be an eigenvector with eigenvalue $\lambda$; then $v = s(s(v)) = s (\lambda v) = \lambda^2 v$. Then $v (\lambda^2 - 1) = 0$; therefore, since $v$ is nonzero, we have $\lambda^2 - 1 = 0$ and thus $\lambda = \pm 1$.

In a bit more generality, if $s$ is any linear operator, $P$ is a polynomial, $P(s) = 0$, and $\lambda$ is an eigenvalue, then $P(\lambda) = 0$. The $P$ in this case is $P(x) = x^2 - 1$.