Pauli matrices

The Pauli matrices are the complex two-by-two matricesEach is a traceless matrix, a Hermitian matrix and a unitary matrix. Their eigenvalues are $+1,-1$, and direct matrix multiplication gives ${\sigma }_x{\sigma }_y=i{\sigma }_z$, with cyclic permutations and a minus sign for reversed order. This is the Pauli matrix multiplication law.

Together with the identity matrix, they form the Pauli operator basis of all complex two-by-two matrices. For spin one-half, the angular momentum operators are $S_j=\hslash {\sigma }_j/2$, so a spin measurement along a unit vector $n$ has outcomes $\pm \hslash /2$. For example, ${\sigma }_z$ has the standard basis as its eigenbasis, whereas ${\sigma }_x$ has normalized eigenvectors $(1,\pm 1{)}^T/\sqrt{2}$. This makes changing the measurement axis an explicit change of basis.^[ref]