Find eigenvalues and eigenvectors of a linear transformation

Question

Let be $V=\mathbb{R}^4$ with the standard scalar product, and $F:V\rightarrow V$ symmetric (its matrix representation satisfies $A^t=A$) and biunivocal and $F^3=F$ (so $F(F(F(v)))=F(v)$). The set of vectors $\vec{v} \in V$ such that $F(\vec{v})=\vec{v}$ is $$S=\{(x_1,x_2,x_3.x_4) \:\:| \: x_1=x_2 ,\: x_3=x_4\}$$

Find eigenvalues and eigenvectors of $F$.

Can you give me a hint about how to start?

Answer 1

We can make the following observations:

• Since $F^3 = F$, we can multiply both sides by $F^{-1}$ to find that $F^2 = I$ (where $I$ denotes the identity matrix).

• The only eigenvalues $F$ can have are $1$ and $-1$.

• $S$ is the eigenspace of $F$ associated with $1$.

• Because $F$ is symmetric and our earlier statement about eigenvalues, the elements of $S^\perp$ must be eigenvectors associated with $-1$.