= Antipodal eigenprojectors of a qubit density matrix
{title2=$P_\pm=(I\pm\widehat n\cdot\sigma)/2$}
For a qubit <density matrix> $\rho=(I+n\cdot\sigma)/2$ with nonzero <Bloch vector>, $\widehat n=n/\|n\|$ and the multiplication rule for the <Pauli matrices> give these orthogonal rank-one <spectral projectors>, with eigenvalues $(1\pm\|n\|)/2$. Their <Bloch vectors> are opposite unit vectors $\pm\widehat n$. At $n=0$ the state is maximally mixed and the direction is undefined; any orthonormal eigenbasis gives an antipodal pair, but no pair is distinguished by the state.
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