<s>[INST] How to determine the eigenvectors for this matrix I have the matrix $$\left( \begin{array}{ccc}
-\alpha & \beta \\
\beta/K &  -\alpha/K
\end{array} \right)$$ for which the eigenvalues are $$\lambda_{1,2}=-\dfrac{\alpha}{2}-\dfrac{\beta}{2K}\pm\dfrac{\sqrt{(K\alpha-\beta)^2+4K\alpha\beta }}{2K}$$ I have a problem with finding the eigenvectors. It seems that the eigenvectors are zero vectors but then eigenvectors cannot be zero. What am I missing?
 [/INST] To find the eigenvalues of a matrix $A$, we solve $\mathrm{det}(A-\lambda I)=0$ for $\lambda$. In this case:
$\begin{align}
\left|\begin{array}{cc} -\alpha-\lambda & \beta \\ \frac{\beta}{K} &  \frac{-\alpha}{K}-\lambda \end{array}\right| &= (-\alpha-\lambda)\left(\frac{-\alpha}{K}-\lambda\right)-\frac{\beta^2}{K} \\
&=\frac{\alpha^2}{K}+\left(\alpha+\frac{\alpha}{K}\right)\lambda + \lambda^2 - \frac{\beta^2}{K} \\
&=\lambda^2 + \left(\alpha+\frac{\alpha}{K}\right)\lambda + \frac{\alpha^2-\beta^2}{K} = 0,
\end{align}$
implies that our eigenvalues are:
$\begin{align}
\lambda_{1,2} &= -\frac{\alpha+\frac{\alpha}{K}}{2} \pm \frac{\sqrt{\left(\alpha+\frac{\alpha}{K}\right)^2 - 4\frac{\alpha^2-\beta^2}{K}}}{2} \\
&=-\frac{K\alpha+\alpha}{2K} \pm \frac{\sqrt{(K\alpha+\alpha)^2 - 4K(\alpha^2-\beta^2)}}{2K} \\
&=-\frac{(K+1)\alpha}{2K} \pm \frac{\sqrt{\left((K+1)^2-4K\right)\alpha^2 + 4K\beta^2}}{2K} \\
&=-\frac{(K+1)\alpha}{2K} \pm \frac{\sqrt{(K-1)^2\alpha^2 + 4K\beta^2}}{2K}.
\end{align}$
Now, an eigenvector corresponding to the eigenvalue $\lambda_i$ is in the nullspace of $\left(\begin{array}{cc} -\alpha-\lambda_i & \beta \\ \frac{\beta}{K} &  \frac{-\alpha}{K}-\lambda_i \end{array}\right).$
It appears that the vector $\bf{v}_i=\left(\begin{array}{c} \beta \\ \alpha + \lambda_i \end{array}\right)$ would work.
</s>