 Traceless symmetric two-by-two matrix

For real $a,b$, the displayed symmetric matrix satisfies $M^2=(a^2+b^2)I$. If $s=\sqrt{a^2+b^2}>0$, choose $\theta$ with $\cos 2\theta =a/s$ and $\sin 2\theta =b/s$. The eigenvectors $(\cos \theta ,\sin \theta )$ and $(-\sin \theta ,\cos \theta )$ are orthonormal and have eigenvalues $s$ and $-s$. The half-angle description works even when $b=0$, avoiding an artificial division by an off-diagonal entry. If $s=0$, the matrix is zero and every direction is an eigenvector. The normalized matrix $M/s$ is reflection in the axis at angle $\theta$.