Solution
= Solution
{c}
Using the convention $\chi_T(x)=\det(xI-A)$, the <characteristic polynomial> is
$$
\boxed{\chi_T(x)=(x-1)\det\begin{pmatrix}x&1\\-1&x\end{pmatrix}=(x-1)(x^2+1).}
$$
Since $x^2+1>0$ for every $x\in\mathbb R$, the only <eigenvalue> over the <real numbers> is $1$. Solving $(A-I)v=0$ gives
$$
\boxed{E_1=\ker(A-I)=\operatorname{span}\{(0,0,1)\}.}
$$
Thus the only <eigenspace> over the <real numbers> is the axis of the <rotation>.