= Pauli matrices
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{title2=$\sigma_x,\sigma_y,\sigma_z$}
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= Pauli matrix
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The Pauli matrices are the complex two-by-two <matrices>
$$
\sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
\sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\qquad
\sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
$$
Each 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 $S_j=\hbar\sigma_j/2$, so a <spin measurement> along a <unit vector> $n$ has outcomes $\pm\hbar/2$. For example, $\sigma_z$ has the <standard basis> as its <eigenbasis>, whereas $\sigma_x$ has normalized <eigenvectors> $(1,\pm1)^T/\sqrt2$. This makes changing the measurement axis an explicit <change of basis>.<The Feynman Lectures on Physics, Volume III by Richard Feynman and others>[Chapter 11, section 11-1, equations 11.5-11.10 and Table 11-1]{ref}
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