<s>[INST] Eigenvectors of $\begin{pmatrix}6&2\\-10&-1\end{pmatrix}$ (linear equations with complex numbers) I want to compute the eigenvectors of the matrix
$$\begin{pmatrix}6&2\\-10&-1\end{pmatrix}$$
and thus far I got the eigenvalues $\lambda_{1,2}=\frac{1}{2}(5\pm\sqrt{31}i)$. However solving for example the system
$$\begin{pmatrix}6-\frac{1}{2}(5+\sqrt{31}i)&2\\-10&-1-\frac{1}{2}(5+\sqrt{31}i)\end{pmatrix}\cdot v=0$$
for the first eigenvector was pretty difficult and I couldn't come to a proper solution whatsoever. Do you have suggestions on how to simplify the system?
 [/INST] We have:
$$\begin{pmatrix}6-\frac{1}{2}(5+\sqrt{31}i)&2\\-10&-1-\frac{1}{2}(5+\sqrt{31}i)\end{pmatrix}\cdot v=0$$
Lets simplify:
$$\begin{pmatrix}\frac{1}{2}(7-\sqrt{31}i)&2\\-10&-\frac{1}{2}(7+\sqrt{31}i)\end{pmatrix}\cdot v=0$$
Divide the first row by $\frac{1}{2}(7-\sqrt{31}i)$ and the second row by $-10$:
$$\begin{pmatrix}1 & \dfrac{4}{(7-\sqrt{31}i)}\\1&\dfrac{1}{20}(7+\sqrt{31}i)\end{pmatrix}\cdot v=0$$
Multiply $a_{12}$ by the conjugate:
$$\begin{pmatrix}1 & \dfrac{1}{20}(7+\sqrt{31}i)\\1&\dfrac{1}{20}(7+\sqrt{31}i)\end{pmatrix}\cdot v=0$$
$R_2 - R_1 \rightarrow R_2$:
$$\begin{pmatrix}1 & \dfrac{1}{20}(7+\sqrt{31}i)\\0&0\end{pmatrix}\cdot v=0$$
So:
$$v_1 = \left(-\dfrac{1}{20}(7+\sqrt{31}i), ~~1 \right)$$
Given that the eigenvalues are complex cojugates, so too are the eigenvectors, so you already know $v_2$.
However, you might want to give that one a go and make sure you can verify that.
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