Solution
= Solution
For an <eigenvalue> $t\in\{\varphi,\psi\}$, the equation $A(x,y)^T=t(x,y)^T$ gives $x=ty$ and $(t+1)y=t^2y$. The second identity holds because $t^2=t+1$. Taking $y=1$ gives the <eigenvectors>
$$
\boxed{v_\varphi=\begin{pmatrix}\varphi\\1\end{pmatrix},\qquad v_\psi=\begin{pmatrix}\psi\\1\end{pmatrix}.}
$$
Each associated <eigenspace> consists of all scalar multiples of its displayed vector. A zero vector is excluded when speaking of an <eigenvector>.