= Hermitian two-by-two matrix in the Pauli basis
{c}
{title2=$M=dI+\mathbf b\cdot\boldsymbol\sigma$}
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 $M=\begin{pmatrix}u&w\\w^*&z\end{pmatrix}$, take $d=(u+z)/2$, $b_z=(u-z)/2$, $b_x=\operatorname{Re}w$ and $b_y=-\operatorname{Im}w$. Alternatively, $d=\operatorname{tr}M/2$ and $b_i=\operatorname{tr}(M\sigma_i)/2$ follow from the trace orthogonality of the <Pauli matrices> and the <identity matrix>.
The <Pauli matrix multiplication law> gives $(\mathbf b\cdot\boldsymbol\sigma)^2=|\mathbf b|^2I$, hence the two <eigenvalues> are $d\pm|\mathbf b|$. For $\mathbf b\ne0$, the corresponding pure-state <Bloch vectors> are $\pm\mathbf b/|\mathbf b|$.
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