# Pauli matrices

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In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries.

{\begin{aligned}\sigma _{1}=\sigma _{\mathrm {x} }&={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\\\sigma _{2}=\sigma _{\mathrm {y} }&={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\\\sigma _{3}=\sigma _{\mathrm {z} }&={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}\\\end{aligned}}  Wolfgang Pauli (1900–1958), ca. 1924. Pauli received the Nobel Prize in physics in 1945, nominated by Albert Einstein, for the Pauli exclusion principle.

These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left).

Each Pauli matrix is Hermitian, and together with the identity matrix I (sometimes considered as the zeroth Pauli matrix σ0), the Pauli matrices form a basis for the real vector space of 2 × 2 Hermitian matrices. This means that any 2 × 2 Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers.

Hermitian operators represent observables in quantum mechanics, so the Pauli matrices span the space of observables of the complex 2-dimensional Hilbert space. In the context of Pauli's work, σk represents the observable corresponding to spin along the kth coordinate axis in three-dimensional Euclidean space $\mathbb {R} ^{3}.$ The Pauli matrices (after multiplication by i to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: the matrices 1, 2, 3 form a basis for the real Lie algebra ${\mathfrak {su}}(2)$ , which exponentiates to the special unitary group SU(2).[lower-alpha 1] The algebra generated by the three matrices σ1, σ2, σ3 is isomorphic to the Clifford algebra of $\mathbb {R} ^{3}$ , and the (unital associative) algebra generated by 1, 2, 3 is effectively identical (isomorphic) to that of quaternions ($\mathbb {H}$ ).