Real eigenvalues of a matrix

Idea. A real $2\times2$ matrix has real eigenvalues iff its characteristic polynomial has nonnegative discriminant. In the $(z,w)$ coordinates, this discriminant is a difference of two squares.

Step 1: trace and determinant. $\operatorname{tr}A=a+d=2\operatorname{Re}z$ and $\det A=|z|^2-|w|^2$ (previous problems).

Step 2: discriminant. The characteristic polynomial $t^2-(\operatorname{tr}A)t+\det A$ has discriminant
$$\Delta=(\operatorname{tr}A)^2-4\det A=4(\operatorname{Re}z)^2-4|z|^2+4|w|^2=4\bigl(|w|^2-(\operatorname{Im}z)^2\bigr),$$
since $|z|^2=(\operatorname{Re}z)^2+(\operatorname{Im}z)^2$.

Step 3: conclusion. $A$ has real eigenvalues iff $\Delta\ge0$ iff $|w|\ge|\operatorname{Im}z|$. In that case the eigenvalues are
$$\lambda_\pm=\frac{\operatorname{tr}A\pm\sqrt\Delta}2=\operatorname{Re}z\pm\sqrt{|w|^2-(\operatorname{Im}z)^2}.$$
If $|w|<|\operatorname{Im}z|$ the eigenvalues are the non-real pair $\operatorname{Re}z\pm i\sqrt{(\operatorname{Im}z)^2-|w|^2}$.

Translation to entries. $\operatorname{Im}z=\frac{c-b}2$ and $|w|^2=\frac{(a-d)^2+(b+c)^2}4$, so
$$|w|^2-(\operatorname{Im}z)^2=\frac{(a-d)^2+(b+c)^2-(c-b)^2}4=\frac{(a-d)^2+4bc}4,$$
which is the familiar discriminant $(a-d)^2+4bc\ge0$.

Check. Rotation $R(\theta)$: $z=e^{i\theta}$, $w=0$, so $|w|=0<|\sin\theta|=|\operatorname{Im}z|$ unless $\theta\in\pi\mathbb{Z}$; real eigenvalues only for $R(0)=I$ and $R(\pi)=-I$. Reflection $\mathrm{diag}(1,-1)$: $z=0$, $w=1$, eigenvalues $0\pm\sqrt1=\pm1$. Numerically, the formula $\operatorname{Re}z\pm\sqrt{|w|^2-(\operatorname{Im}z)^2}$ matches `numpy` eigenvalues for $200$ random matrices.

Remark. The set of matrices with a real eigenvalue is $\{|w|\ge|\operatorname{Im}z|\}$: it is closed, with interior $\{|w|>|\operatorname{Im}z|\}$ and boundary $\{|w|=|\operatorname{Im}z|\}$ (matrices with a repeated real eigenvalue). This recovers the description of the interior in terms of $\operatorname{tr}^2>4\det$.

Lesson. When a discriminant condition appears, look for coordinates in which it factors or becomes a comparison of two moduli.