# Pauli matrices

The **Pauli matrices** are a set of three 2 × 2 complex Hermitian, traceless matrices, denoted σ₁, σ₂, σ₃, or σx, σy, σz, that describe the spin of spin-1/2 particles and serve as the fundamental representation of the [Lie algebra](https://www.edgepedia.org/lie-algebra) of the group SU(2).<sup>[1](https://quantummechanics.ucsd.edu/ph130a/130_notes/node28.html)</sup><sup> • </sup><sup>[2](https://www.phy.olemiss.edu/~luca/Topics/s/SU2.html)</sup>  In the eigenbasis of σz they are written<sup>[3](https://www2.physics.siu.edu/~shajesh/rp/files/QM-lnotes.pdf)</sup>

\[ \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. \]

| Key fact | Detail |
|---|---|
| Definition | Three 2 × 2 Hermitian, traceless matrices σx, σy, σz; together with the identity they form a basis for all 2 × 2 matrices<sup>[1](https://quantummechanics.ucsd.edu/ph130a/130_notes/node28.html)</sup> |
| Product identity | σᵢσⱼ = δᵢⱼ + i εᵢⱼₖ σₖ, with i, j, k in {1, 2, 3}<sup>[3](https://www2.physics.siu.edu/~shajesh/rp/files/QM-lnotes.pdf)</sup> |
| Commutators | σxσy = −σyσx = iσz, and cyclic permutations<sup>[4](https://math.mit.edu/~shor/18.435/oldlectures/lecture3.html)</sup> |
| Anticommutators | {σᵢ, σⱼ} = 2δᵢⱼ; each matrix squares to the identity, σx² = σy² = σz² = 1<sup>[1](https://quantummechanics.ucsd.edu/ph130a/130_notes/node28.html)</sup><sup> • </sup><sup>[4](https://math.mit.edu/~shor/18.435/oldlectures/lecture3.html)</sup>  |
| Physical role | Spin operators for a spin-1/2 particle are Sᵢ = (ħ/2)σᵢ<sup>[1](https://quantummechanics.ucsd.edu/ph130a/130_notes/node28.html)</sup> |
| Origin | Introduced by Wolfgang Pauli in "Zur Quantenmechanik des magnetischen Elektrons", published in May 1927 in *Zeitschrift für Physik* 43, pp. 601–623<sup>[5](https://library.ethz.ch/en/collections-and-archives/platforms/virtual-exhibitions/wolfgang-pauli-and-modern-physics/the-old-testament-and-the-pauli-matrices.html)</sup><sup> • </sup><sup>[6](https://cir.nii.ac.jp/crid/1362262945346727552)</sup>  |
| Generalization | The eight 3 × 3 Gell-Mann matrices play the analogous role for SU(3)<sup>[7](https://web2.ph.utexas.edu/~vadim/Classes/2024f-qft/gauge.pdf)</sup> |

## Algebraic properties

The defining identity combines the commutator and anticommutator in one product rule,<sup>[3](https://www2.physics.siu.edu/~shajesh/rp/files/QM-lnotes.pdf)</sup>

\[ \sigma_i \sigma_j = \delta_{ij} \, 1 + i \sum_k \epsilon_{ijk} \, \sigma_k, \]

where δᵢⱼ is the [Kronecker delta](https://www.edgepedia.org/kronecker-delta) and εᵢⱼₖ the [Levi-Civita symbol](https://www.edgepedia.org/levi-civita-symbol). Taking the antisymmetric part gives the commutation relations<sup>[4](https://math.mit.edu/~shor/18.435/oldlectures/lecture3.html)</sup>

\[ [\sigma_i, \sigma_j] = 2 i \, \epsilon_{ijk} \, \sigma_k, \]

explicitly σxσy = −σyσx = iσz, σyσz = −σzσy = iσx, and σzσx = −σxσz = iσy.<sup>[4](https://math.mit.edu/~shor/18.435/oldlectures/lecture3.html)</sup> Taking the symmetric part gives the anticommutation relations {σᵢ, σⱼ} = 2δᵢⱼ, so distinct Pauli matrices anticommute and each squares to the identity.<sup>[1](https://quantummechanics.ucsd.edu/ph130a/130_notes/node28.html)</sup><sup> • </sup><sup>[4](https://math.mit.edu/~shor/18.435/oldlectures/lecture3.html)</sup>  These relations are independent of the arbitrary phases in the eigenvector choices of a representation, so no measurable quantity depends on that choice.<sup>[3](https://www2.physics.siu.edu/~shajesh/rp/files/QM-lnotes.pdf)</sup>

As a basis, the three Pauli matrices and the identity span the space of 2 × 2 matrices: any 2 × 2 matrix is a linear combination of them, and any 2 × 2 [Hermitian matrix](https://www.edgepedia.org/hermitian-matrix) is a real-weighted combination of the four.<sup>[1](https://quantummechanics.ucsd.edu/ph130a/130_notes/node28.html)</sup><sup> • </sup><sup>[8](https://books.physics.oregonstate.edu/LinAlg/matrixex.html)</sup>  Each Pauli matrix is traceless and Hermitian, with eigenvalues +1 and −1.<sup>[1](https://quantummechanics.ucsd.edu/ph130a/130_notes/node28.html)</sup><sup> • </sup><sup>[3](https://www2.physics.siu.edu/~shajesh/rp/files/QM-lnotes.pdf)</sup>

## Spin-1/2 and rotations

For a spin-1/2 particle the spin operators are Sᵢ = (ħ/2)σᵢ, so the two eigenvalues of each spin component are ±ħ/2.<sup>[1](https://quantummechanics.ucsd.edu/ph130a/130_notes/node28.html)</sup> A spin pointing along an arbitrary unit direction r̂ = (sin θ cos φ, sin θ sin φ, cos θ) is represented by the Hermitian matrix σ · r̂, whose eigenvalues are again ±1.<sup>[3](https://www2.physics.siu.edu/~shajesh/rp/files/QM-lnotes.pdf)</sup><sup> • </sup><sup>[4](https://math.mit.edu/~shor/18.435/oldlectures/lecture3.html)</sup>

The commutation relations make the Pauli matrices a basis of the Lie algebra su(2) in its fundamental (two-dimensional) representation; with the convention σᵢ = (i/2)τᵢ they generate the algebra of SU(2).<sup>[2](https://www.phy.olemiss.edu/~luca/Topics/s/SU2.html)</sup> Exponentiating linear combinations of Pauli matrices yields SU(2) matrices.<sup>[9](https://www.math.columbia.edu/~woit/QM/fall-course.pdf)</sup> Higher-dimensional representations exist for every half-integer spin: three (2n+1) × (2n+1) Hermitian matrices J₁, J₂, J₃ with the same commutation relations, the Pauli matrices corresponding to n = 1/2.<sup>[10](https://www.hep.ucl.ac.uk/~lang/lhl_aQFT.pdf)</sup> The Pauli matrices also satisfy a [Clifford algebra](https://www.edgepedia.org/clifford-algebra): σ₁, σ₂, σ₃ are [gamma matrices](https://www.edgepedia.org/gamma-matrices) for the Clifford algebra C(0,3), and together with the 2 × 2 identity they generate an eight-dimensional real algebra isomorphic to C(0,3).<sup>[11](https://www.sciencedirect.com/topics/mathematics/pauli-matrix)</sup>

## History

[Wolfgang Pauli](https://www.edgepedia.org/wolfgang-pauli) introduced the matrices in the paper "Zur Quantenmechanik des magnetischen Elektrons" (On the quantum mechanics of the magnetic electron), published in May 1927 in *Zeitschrift für Physik* volume 43, pages 601–623.<sup>[5](https://library.ethz.ch/en/collections-and-archives/platforms/virtual-exhibitions/wolfgang-pauli-and-modern-physics/the-old-testament-and-the-pauli-matrices.html)</sup><sup> • </sup><sup>[6](https://cir.nii.ac.jp/crid/1362262945346727552)</sup>  The paper incorporated the electron spin hypothesis of Goudsmit and Uhlenbeck into quantum mechanics using two-component wave functions, splitting an electron's eigenfunction into two spatial functions ψα and ψβ.<sup>[12](https://oamonitor.ireland.openaire.eu/rpo/rcsi/search/publication?pid=10.1007%2F978-3-322-90270-2_32)</sup> Pauli had not been among the originators of matrix mechanics in 1925; his 1926 chapter on quantum theory in the *Handbuch der Physik* was so clear that physicists jokingly called it the "Old Testament".<sup>[5](https://library.ethz.ch/en/collections-and-archives/platforms/virtual-exhibitions/wolfgang-pauli-and-modern-physics/the-old-testament-and-the-pauli-matrices.html)</sup> Pauli received the 1945 [Nobel Prize in Physics](https://www.edgepedia.org/nobel-prize-in-physics) for the discovery of the Exclusion Principle, also called the Pauli Principle.<sup>[13](https://www.livescience.com/16362-nobel-prize-physics-list.html)</sup>

## Generalizations and applications

The **Gell-Mann matrices** are the 3 × 3 analog for SU(3): eight Hermitian, traceless matrices that, like the Pauli matrices, generate a Lie algebra of unitary transformations.<sup>[7](https://web2.ph.utexas.edu/~vadim/Classes/2024f-qft/gauge.pdf)</sup><sup> • </sup><sup>[14](https://mathworld.wolfram.com/Gell-MannMatrix.html)</sup>  They differ in algebraic detail: unlike the Pauli matrices, they do not all anticommute with each other and do not all square to unit matrices.<sup>[7](https://web2.ph.utexas.edu/~vadim/Classes/2024f-qft/gauge.pdf)</sup> In gauge theory, the SU(2) generators are written in terms of the three Pauli matrices, for example T_z = σz/2.<sup>[15](https://www.tpi.uni-jena.de/~floerchinger/qft2/lecture09/)</sup>

In quantum computing the Pauli matrices appear directly as single-qubit gates. The X gate, whose matrix representation is σx, applies a π rotation around the x axis of the [Bloch sphere](https://www.edgepedia.org/bloch-sphere) and maps |0⟩ to |1⟩ and |1⟩ to |0⟩, acting like a classical NOT gate.<sup>[16](https://quantum.cloud.ibm.com/learning/en/courses/utility-scale-quantum-computing/bits-gates-and-circuits)</sup> Products of Pauli matrices on multiple qubits, called Pauli strings, are used to represent and evolve operators in quantum simulation software; IBM's Pauli-propagation tools compute operator evolution as sums over Pauli terms.<sup>[17](https://quantum.cloud.ibm.com/docs/en/api/pauli-prop/propagation)</sup>

## References

1. [Spin 1/2 and other 2 State Systems, UCSD Physics 130A notes](https://quantummechanics.ucsd.edu/ph130a/130_notes/node28.html)
2. [Topics: SU(2) Group, University of Mississippi](https://www.phy.olemiss.edu/~luca/Topics/s/SU2.html)
3. [Notes on Quantum Mechanics, Southern Illinois University (Shajesh)](https://www2.physics.siu.edu/~shajesh/rp/files/QM-lnotes.pdf)
4. [18.435 Quantum Computation, Lecture 3, MIT (Peter Shor)](https://math.mit.edu/~shor/18.435/oldlectures/lecture3.html)
5. [The "Old Testament" and the Pauli matrices, ETH Library](https://library.ethz.ch/en/collections-and-archives/platforms/virtual-exhibitions/wolfgang-pauli-and-modern-physics/the-old-testament-and-the-pauli-matrices.html)
6. [Zur Quantenmechanik des magnetischen Elektrons, CiNii Research](https://cir.nii.ac.jp/crid/1362262945346727552)
7. [Gauge Theories lecture notes, University of Texas at Austin (Vadim Kazakov)](https://web2.ph.utexas.edu/~vadim/Classes/2024f-qft/gauge.pdf)
8. [Matrix Examples, Oregon State University Linear Algebra](https://books.physics.oregonstate.edu/LinAlg/matrixex.html)
9. [Quantum Theory, Groups and Representations (Peter Woit), Columbia University](https://www.math.columbia.edu/~woit/QM/fall-course.pdf)
10. [Quantum Field Theory lecture notes, UCL](https://www.hep.ucl.ac.uk/~lang/lhl_aQFT.pdf)
11. [Pauli Matrix, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/pauli-matrix)
12. [Zur Quantenmechanik des magnetischen Elektrons, OpenAIRE](https://oamonitor.ireland.openaire.eu/rpo/rcsi/search/publication?pid=10.1007%2F978-3-322-90270-2_32)
13. [Nobel Prize in Physics: 1901–Present, Live Science](https://www.livescience.com/16362-nobel-prize-physics-list.html)
14. [Gell-Mann Matrix, Wolfram MathWorld](https://mathworld.wolfram.com/Gell-MannMatrix.html)
15. [Quantum field theory 2, lecture 09, University of Jena (Floerchinger group)](https://www.tpi.uni-jena.de/~floerchinger/qft2/lecture09/)
16. [Bits, gates, and circuits, IBM Quantum Learning](https://quantum.cloud.ibm.com/learning/en/courses/utility-scale-quantum-computing/bits-gates-and-circuits)
17. [Pauli propagation API, IBM Quantum Documentation](https://quantum.cloud.ibm.com/docs/en/api/pauli-prop/propagation)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Linear and multilinear algebra › Matrix theory › Structured and special matrix classes*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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