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Take note that $\vec \sigma$, a "vector of matrices" is only used as a convenient short-hand notation for the sums that appear in the equation. For example, $\vec \sigma \vec p = \sigma_1 p_1 + \sigma_2 p_2 + \sigma_3 p_3. $
The Pauli equation is the correct non-relativistic equation to describe spin $1/2$ particles.
The Pauli equation is the non-relativistic limit of the Dirac equation.