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In physics, the Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices.^{[1]} Usually indicated by the Greek letter "sigma" (σ), they are occasionally denoted with a "tau" (τ) when used in connection with isospin symmetries. They are:
The name refers to Wolfgang Pauli.
The real (hence also, complex) subalgebra generated by the σ_{i} (that is, the set of real or complex linear combinations of all the elements which can be built up as products of Pauli matrices) is the full set M_{2}(C) of complex 2×2 matrices. The σ_{i} can also be seen as generating the real Clifford algebra of the real quadratic form with signature (3,0): this shows that this Clifford algebra Cℓ_{3,0}(R) is isomorphic to M_{2}(C), with the Pauli matrices providing an explicit isomorphism. (In particular, the Pauli matrices define a faithful representation of the real Clifford algebra Cℓ_{3,0}(R) on the complex vector space C^{2} of dimension 2.)
Contents
Algebraic properties
where I is the identity matrix, i.e. the matrices are involutory.
From above we can deduce that the eigenvalues of each σ_{i} are ±1.
 Together with the identity matrix I (which is sometimes written as σ_{0}), the Pauli matrices form an orthogonal basis, in the sense of HilbertSchmidt, for the real Hilbert space of 2 × 2 complex Hermitian matrices, or the complex Hilbert space of all 2 × 2 matrices.
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