Hermitian two-by-two matrix in the Pauli basis ($M=dI+\mathbf{b}\cdot \mathbf{\sigma }$)

Every Hermitian matrix of size two can be written uniquely as $M=dI+b_x{\sigma }_x+b_y{\sigma }_y+b_z{\sigma }_z$ with real coefficients. For $M=\left(\begin{array}{cc}u& w\\ w^{\ast }& z\end{array}\right)$, take $d=(u+z)/2$, $b_z=(u-z)/2$, $b_x=\mathrm{Re}w$ and $b_y=-\mathrm{Im}w$. Alternatively, $d=\mathrm{tr}M/2$ and $b_i=\mathrm{tr}(M{\sigma }_i)/2$ follow from the trace orthogonality of the Pauli matrices and the identity matrix.

The Pauli matrix multiplication law gives $(\mathbf{b}\cdot \mathbf{\sigma }{)}^2=|\mathbf{b}{|}^{2}I$, hence the two eigenvalues are $d\pm |\mathbf{b}|$. For $\mathbf{b}\ne 0$, the corresponding pure-state Bloch vectors are $\pm \mathbf{b}/|\mathbf{b}|$.