Pairs of eigenvectors that are not multiple of each other I have the following matrix
\begin{align}
J
&= \begin{bmatrix}
0 & \lambda a \\
a & \delta \\
\end{bmatrix}
\end{align}
For which I find two eigenvalues ($e_-$ and $e_+$):
\begin{align}
\det \left( J - e I \right) &= 0 \\
\det \left( 
\begin{bmatrix}
-e & \lambda a  \\
a & \delta -e \\
\end{bmatrix} \right) &= 0 \nonumber \\
e^2 - \delta e - \lambda a^2 &= 0 \nonumber \\
e_\pm = \frac{\delta \pm \sqrt{\delta^2 + 4 \lambda a^2}}{2}
\end{align}
Then I find the eigenvectors (denoted with v):
\begin{align}
\left( J - e_\pm I \right) v &= 0 \\
\begin{bmatrix}
- e_\pm & \lambda a \\
a & \delta - e_\pm \\
\end{bmatrix} 
\begin{bmatrix}
v^1 \\
v^2 \\
\end{bmatrix} &=
\begin{bmatrix}
0 \\
0 \\
\end{bmatrix} \nonumber \\
\begin{bmatrix}
- e_\pm v^1 + \lambda a v^2 \\
a v^1 + (\delta - e_\pm) v^2 \\
\end{bmatrix} 
&=
\begin{bmatrix}
0 \\
0 \\
\end{bmatrix} \nonumber
\end{align}
For the eigenvalue $e_+$:
\begin{align}
v^2 &= \frac{e_+}{\lambda a} v^1 
\end{align}
\begin{align}
v^1 &= \frac{e_+ - \delta}{a} v^2  
\end{align}
For the eigenvalue $e_-$:
\begin{align}
v^2 &= \frac{e_-}{\lambda a} v^1
\end{align}
\begin{align}
v^1 &= \frac{e_- - \delta}{a} v^2  
\end{align}
Therefore, setting $v^1 = 1$:
\begin{align}
\left\{ e_+ \ ; \ [ 1, \frac{e_+}{\lambda a}]^\top \right\} \quad \left\{ e_- \ ; \ [ 1, \frac{e_-}{\lambda a}]^\top \right\}
\end{align}
Setting $v^2 = 1$:
\begin{align}
\left\{ e_+ \ ; \ [ \frac{e_+ - \delta}{a}, 1]^\top \right\} \quad \left\{ e_- \ ; \ [ \frac{e_- - \delta}{a}, 1]^\top \right\}
\end{align}
Could you clarify why I obtain two pairs of eigenvectors that are not multiple of each other? How can the number of eigenvectors be greater than the dimension of the matrix?

They are scalar multiples of each other:
$$
\left(\frac{e_+-\delta}{a},1\right)
=\frac{e_+-\delta}{a}\left(1,\frac{e_+}{\lambda a}\right).
$$
Note that $\dfrac{e_+-\delta}{a}\dfrac{e_+}{\lambda a}=1$ because $e_+$ is a root of the characteristic equation $x^2-\delta x-\lambda a^2=0$ of $J$.
