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Definition 2024

Pauli matrix ‎(plural Pauli matrixes or Pauli matrices)

(mathematics) Any one of three complex matrices that arise in Wolfgang Pauli's treatment of spin in quantum mechanics.

The Pauli matrices satisfy the equation $\sigma _{1}^{2}=\sigma _{2}^{2}=\sigma _{3}^{2}=-i\sigma _{1}\sigma _{2}\sigma _{3}=I$ which is reminiscent of Hamilton's fundamental formula for quaternion multiplication.

The first Pauli matrix is like a reflection about the "y=x" line. The third Pauli matrix is like a reflection about the "x axis". The second Pauli matrix is like a 90° counterclockwise rotation and scalar multiplication by the imaginary unit (which rotates the two complex components by 90° counterclockwise in their own "phasor spaces", as it were).