Let $A: \mathbb{R}^6 \rightarrow \mathbb{R}^6$ be linear transformation, $A^{26} = I$.

Question

Let $A: \mathbb{R}^6 \rightarrow \mathbb{R}^6$ be linear transformation, $A^{26} = I$. Find linear spaces $V_1, V_2, V_3$, such that: $\mathbb{R}^6 = V_1 \oplus V_2 \oplus V_3$, dim $V_1$ = dim $V_2$ = dim $V_3$ and $AV_1 \subset V_1, AV_2 \subset V_2, AV_3 \subset V_3$. Can you help me?

I see that $\mu_1\mu_2\ldots\mu_6 = \textrm{det}A$ $\in \{-1,1\}$ and $\mu_i \in \{e^{\frac{2k\pi i}{26}}\mid k = 0,1,\ldots,25\}$. I need to know more about $\sigma (A)$. Is it possible to show $1$ or $-1$ has to be in the spectrum?

Answer 1

If $A$ has a complex eigenvalue $\lambda=a+bi (b\neq 0)$, then the corresponding (complex) eigenvector is of the form $X+iY$ where $X,Y$ have real coordinates, and it is easy to see that $X$ and $Y$ must be linearly independent. Then the subspace $V_{\lambda}={\textsf{span}}(X,Y)$ is two-dimensional and invariant by $A$. The other eigenvalues are $\pm 1$, so it should be easy for you to finish from here.