First, try to find the eigenvalues with respect to the parameter
$A=\begin{bmatrix}
\alpha&1\\
-1&\alpha\\
\end{bmatrix}$
$det(A-rI)=(\alpha-r)(\alpha-r)+1=0$
$r^2-2{\alpha}r+\alpha^2+1=0$
$r=\frac{2\alpha\pm\sqrt{-4}}{2}$
$r=\alpha\pm2i$ $\color{red}{r_\pm =\alpha \pm i\; V.I.}$
Notice there are always complex eigenvalues, and $\alpha=0$ is critical value since $\alpha=0, \alpha>0, \alpha<0$ have different phase portraits
When $\alpha=0$ , real parts of eigenvalues are 0
When value of $\alpha$ is slightly below 0
Then $\alpha<0$ , real parts of eigenvalues are negative
When value of $\alpha$ is slightly above 0
Then $\alpha>0$ , real parts of eigenvalues are positive
