<s>[INST] 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?
 [/INST] 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 $J$.
</s>