In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary.^{[1]} Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries.
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 σ_{0}), 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 2dimensional Hilbert space. In the context of Pauli's work, σ_{k} represents the observable corresponding to spin along the kth coordinate axis in threedimensional Euclidean space
The Pauli matrices (after multiplication by i to make them antiHermitian) also generate transformations in the sense of Lie algebras: the matrices iσ_{1}, iσ_{2}, iσ_{3} form a basis for the real Lie algebra , which exponentiates to the special unitary group SU(2).^{[a]} The algebra generated by the three matrices σ_{1}, σ_{2}, σ_{3} is isomorphic to the Clifford algebra of ^{[citation needed]}, and the (unital associative) algebra generated by iσ_{1}, iσ_{2}, iσ_{3} is effectively identical (isomorphic) to that of quaternions ().
All three of the Pauli matrices can be compacted into a single expression:
where the solution to i^{2} = 1 is the "imaginary unit", and δ_{jk} is the Kronecker delta, which equals +1 if j = k and 0 otherwise. This expression is useful for "selecting" any one of the matrices numerically by substituting values of j = 1, 2, 3, in turn useful when any of the matrices (but no particular one) is to be used in algebraic manipulations.
The matrices are involutory:
where I is the identity matrix.
The determinants and traces of the Pauli matrices are:
From which, we can deduce that each matrix σ_{jk} has eigenvalues +1 and −1.
With the inclusion of the identity matrix, I (sometimes denoted σ_{0} ), the Pauli matrices form an orthogonal basis (in the sense of Hilbert–Schmidt) of the Hilbert space of real 2 × 2 Hermitian matrices, , and the Hilbert space of all complex 2 × 2 matrices, .
Each of the (Hermitian) Pauli matrices has two eigenvalues, +1 and −1. The corresponding normalized eigenvectors are:
The Pauli vector is defined by^{[b]}
The Pauli vector provides a mapping mechanism from a vector basis to a Pauli matrix basis^{[2]} as follows,
Its normalized eigenvectors are
The Pauli matrices obey the following commutation relations:
and anticommutation relations:
where the structure constant ε_{abc} is the LeviCivita symbol, Einstein summation notation is used, δ_{jk} is the Kronecker delta, and I is the 2 × 2 identity matrix.
For example,
commutators  anticommutators 

Pauli vectors elegantly map these commutation and anticommutation relations to corresponding vector products. Adding the commutator to the anticommutator gives
so that,
Contracting each side of the equation with components of two 3vectors a_{p} and b_{q} (which commute with the Pauli matrices, i.e., a_{p}σ_{q} = σ_{q}a_{p}) for each matrix σ_{q} and vector component a_{p} (and likewise with b_{q}) yields
Finally, translating the index notation for the dot product and cross product results in


(1) 
If i is identified with the pseudoscalar σ_{x}σ_{y}σ_{z} then the right hand side becomes which is also the definition for the product of two vectors in geometric algebra.
If we define the spin operator as J = ħ/2σ, then J satisfies the commutation relation:
The following traces can be derived using the commutation and anticommutation relations.
If the matrix σ_{0} = I is also considered, these relationships become
where Greek indices α, β, γ and μ assume values from {0, x, y, z} and the notation is used to denote the sum over the cyclic permutation of the included indices.
For
one has, for even powers, 2p, p = 0, 1, 2, 3, ...
which can be shown first for the p = 1 case using the anticommutation relations. For convenience, the case p = 0 is taken to be I by convention.
For odd powers, 2q + 1, q = 0, 1, 2, 3, ...
Matrix exponentiating, and using the Taylor series for sine and cosine,
In the last line, the first sum is the cosine, while the second sum is the sine; so, finally,


(2) 
which is analogous to Euler's formula, extended to quaternions.
Note that
while the determinant of the exponential itself is just 1, which makes it the generic group element of SU(2).
A more abstract version of formula (2) for a general 2 × 2 matrix can be found in the article on matrix exponentials. A general version of (2) for an analytic (at a and −a) function is provided by application of Sylvester's formula,^{[3]}
A straightforward application of formula (2) provides a parameterization of the composition law of the group SU(2).^{[c]} One may directly solve for c in
which specifies the generic group multiplication, where, manifestly,
the spherical law of cosines. Given c, then,
Consequently, the composite rotation parameters in this group element (a closed form of the respective BCH expansion in this case) simply amount to^{[4]}
(Of course, when is parallel to , so is , and c = a + b.)
It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation of any angle along any axis :
Taking the dot product of any unit vector with the above formula generates the expression of any single qubit operator under any rotation. For example, it can be shown that .
An alternative notation that is commonly used for the Pauli matrices is to write the vector index k in the superscript, and the matrix indices as subscripts, so that the element in row α and column β of the kth Pauli matrix is σ ^{k}_{αβ}.
In this notation, the completeness relation for the Pauli matrices can be written
The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the Hilbert space of all 2 × 2 complex matrices means that we can express any matrix M as
As noted above, it is common to denote the 2 × 2 unit matrix by σ_{0}, so σ^{0}_{αβ} = δ_{αβ}. The completeness relation can alternatively be expressed as
The fact that any Hermitian complex 2 × 2 matrices can be expressed in terms of the identity matrix and the Pauli matrices also leads to the Bloch sphere representation of 2 × 2 mixed states’ density matrix, (positive semidefinite 2 × 2 matrices with unit trace. This can be seen by first expressing an arbitrary Hermitian matrix as a real linear combination of {σ_{0}, σ_{1}, σ_{2}, σ_{3}} as above, and then imposing the positivesemidefinite and trace 1 conditions.
For a pure state, in polar coordinates,
acts on the state eigenvector with eigenvalue +1, hence it acts like a projection operator.
Let P_{jk} be the transposition (also known as a permutation) between two spins σ_{j} and σ_{k} living in the tensor product space ,
This operator can also be written more explicitly as Dirac's spin exchange operator,
Its eigenvalues are therefore^{[d]} 1 or −1. It may thus be utilized as an interaction term in a Hamiltonian, splitting the energy eigenvalues of its symmetric versus antisymmetric eigenstates.
The group SU(2) is the Lie group of unitary 2 × 2 matrices with unit determinant; its Lie algebra is the set of all 2 × 2 antiHermitian matrices with trace 0. Direct calculation, as above, shows that the Lie algebra is the 3dimensional real algebra spanned by the set {iσ_{k}}. In compact notation,
As a result, each iσ_{j} can be seen as an infinitesimal generator of SU(2). The elements of SU(2) are exponentials of linear combinations of these three generators, and multiply as indicated above in discussing the Pauli vector. Although this suffices to generate SU(2), it is not a proper representation of su(2), as the Pauli eigenvalues are scaled unconventionally. The conventional normalization is λ = 1/2, so that
As SU(2) is a compact group, its Cartan decomposition is trivial.
The Lie algebra is isomorphic to the Lie algebra , which corresponds to the Lie group SO(3), the group of rotations in threedimensional space. In other words, one can say that the iσ_{j} are a realization (and, in fact, the lowestdimensional realization) of infinitesimal rotations in threedimensional space. However, even though and are isomorphic as Lie algebras, SU(2) and SO(3) are not isomorphic as Lie groups. SU(2) is actually a double cover of SO(3), meaning that there is a twotoone group homomorphism from SU(2) to SO(3), see relationship between SO(3) and SU(2).
The real linear span of {I, iσ_{1}, iσ_{2}, iσ_{3}} is isomorphic to the real algebra of quaternions, , represented by the span of the basis vectors The isomorphism from to this set is given by the following map (notice the reversed signs for the Pauli matrices):
Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,^{[5]}
As the set of versors U ⊂ forms a group isomorphic to SU(2), U gives yet another way of describing SU(2). The twotoone homomorphism from SU(2) to SO(3) may be given in terms of the Pauli matrices in this formulation.
In classical mechanics, Pauli matrices are useful in the context of the CayleyKlein parameters.^{[6]} The matrix P corresponding to the position of a point in space is defined in terms of the above Pauli vector matrix,
Consequently, the transformation matrix Q_{θ} for rotations about the xaxis through an angle θ may be written in terms of Pauli matrices and the unit matrix as^{[6]}
Similar expressions follow for general Pauli vector rotations as detailed above.
In quantum mechanics, each Pauli matrix is related to an angular momentum operator that corresponds to an observable describing the spin of a spin 1⁄2 particle, in each of the three spatial directions. As an immediate consequence of the Cartan decomposition mentioned above, iσ_{j} are the generators of a projective representation (spin representation) of the rotation group SO(3) acting on nonrelativistic particles with spin 1⁄2. The states of the particles are represented as twocomponent spinors. In the same way, the Pauli matrices are related to the isospin operator.
An interesting property of spin 1⁄2 particles is that they must be rotated by an angle of 4π in order to return to their original configuration. This is due to the twotoone correspondence between SU(2) and SO(3) mentioned above, and the fact that, although one visualizes spin up/down as the north/south pole on the 2sphere S^{2}, they are actually represented by orthogonal vectors in the two dimensional complex Hilbert space.
For a spin 1⁄2 particle, the spin operator is given by J = ħ/2σ, the fundamental representation of SU(2). By taking Kronecker products of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting spin operators for higher spin systems in three spatial dimensions, for arbitrarily large j, can be calculated using this spin operator and ladder operators. They can be found in Rotation group SO(3) § A note on Lie algebras. The analog formula to the above generalization of Euler's formula for Pauli matrices, the group element in terms of spin matrices, is tractable, but less simple.^{[7]}
Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G_{n} is defined to consist of all nfold tensor products of Pauli matrices.
In relativistic quantum mechanics, the spinors in four dimensions are 4 × 1 (or 1 × 4) matrices. Hence the Pauli matrices or the Sigma matrices operating on these spinors have to be 4 × 4 matrices. They are defined in terms of 2 × 2 Pauli matrices as
It follows from this definition that the matrices have the same algebraic properties as the σ_{k} matrices.
However, relativistic angular momentum is not a threevector, but a second order fourtensor. Hence needs to be replaced by Σ_{μν}, the generator of Lorentz transformations on spinors. By the antisymmetry of angular momentum, the Σ_{μν} are also antisymmetric. Hence there are only six independent matrices.
The first three are the The remaining three, where the Dirac α_{k} matrices are defined as
The relativistic spin matrices Σ_{μν} are written in compact form in terms of commutator of gamma matrices as
In quantum information, singlequbit quantum gates are 2 × 2 unitary matrices. The Pauli matrices are some of the most important singlequbit operations. In that context, the Cartan decomposition given above is called the "Z–Y decomposition of a singlequbit gate". Choosing a different Cartan pair gives a similar "X–Y decomposition of a singlequbit gate".