Pauli Matrices

The Pauli matrices, together with the identity, form an orthogonal basis for all complex $2\times2$ matrices. Their multiplication table compresses much of two-dimensional matrix algebra into ordinary dot and cross products in three coordinates.

This page develops that matrix calculus. The interpretation $S_i=\hbar\sigma_i/2$ as spin components belongs to Pauli Matrices in Symmetry, Angular Momentum, and Spin, while qubit-state geometry belongs to Bloch Sphere Geometry.

Definition

Set $\sigma_0=I_2$. The three Pauli matrices are

$$\begin{aligned} \sigma_x &= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix},\\[6pt] \sigma_y &= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix},\\[6pt] \sigma_z &= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}. \end{aligned}$$

Each $\sigma_i$ is Hermitian and unitary:

$$\sigma_i^\dagger=\sigma_i, \qquad \sigma_i^2=I.$$

It follows that

$$\operatorname{tr}\sigma_i=0, \qquad \det\sigma_i=-1, \qquad \sigma_i^{-1}=\sigma_i.$$

The eigenvalues are $+1$ and $-1$. Thus each Pauli matrix is both an involution and a reflection-like unitary, not a positive operator.

Multiplication Table

For $i,j,k\in\{x,y,z\}$,

$$\sigma_i\sigma_j =\delta_{ij}I +i\sum_k\epsilon_{ijk}\sigma_k.$$

For equal indices this gives $\sigma_i^2=I$. For distinct indices it encodes the cyclic products

$$\begin{aligned} \sigma_x\sigma_y &=i\sigma_z,\\ \sigma_y\sigma_z &=i\sigma_x,\\ \sigma_z\sigma_x &=i\sigma_y, \end{aligned}$$

and reversing an order changes the sign. For example,

$$\sigma_y\sigma_x=-i\sigma_z.$$

The factor of $i$ is essential. It allows the product of two noncommuting Hermitian matrices to be non-Hermitian.

Commutators and Anticommutators

Taking the antisymmetric and symmetric parts of the multiplication rule gives

$$\begin{aligned} [\sigma_i,\sigma_j] &=2i\sum_k\epsilon_{ijk}\sigma_k,\\ \{\sigma_i,\sigma_j\} &=2\delta_{ij}I. \end{aligned}$$

The commutator closes on the three-dimensional span of the Pauli matrices. The anticommutator collapses to the identity. These are the defining relations behind the fundamental representation of the Lie algebra $\mathfrak{su}(2)$ and an irreducible representation of the three-generator complex Clifford relations.

Normalization conventions differ by factors of two. The matrices

$$T_i=\frac{\sigma_i}{2}$$

satisfy

$$[T_i,T_j] =i\sum_k\epsilon_{ijk}T_k.$$

Keep the chosen generators explicit when comparing formulas. See Commutators and Anticommutators and SU(2).

Pauli-Vector Identity

Write

$$\boldsymbol{\sigma} =(\sigma_x,\sigma_y,\sigma_z)$$

and, for $\mathbf a\in\mathbb R^3$,

$$\mathbf a\cdot\boldsymbol{\sigma} =a_x\sigma_x+a_y\sigma_y+a_z\sigma_z.$$

The multiplication table implies

$$(\mathbf a\cdot\boldsymbol{\sigma}) (\mathbf b\cdot\boldsymbol{\sigma}) = (\mathbf a\cdot\mathbf b)I +i(\mathbf a\times\mathbf b) \cdot\boldsymbol{\sigma}.$$

Consequently,

$$\begin{aligned} [ \mathbf a\cdot\boldsymbol{\sigma}, \mathbf b\cdot\boldsymbol{\sigma} ] &= 2i(\mathbf a\times\mathbf b) \cdot\boldsymbol{\sigma},\\ \{ \mathbf a\cdot\boldsymbol{\sigma}, \mathbf b\cdot\boldsymbol{\sigma} \} &= 2(\mathbf a\cdot\mathbf b)I. \end{aligned}$$

Setting $\mathbf b=\mathbf a$ gives

$$(\mathbf a\cdot\boldsymbol{\sigma})^2 =\lVert\mathbf a\rVert^2I.$$

These identities extend bilinearly to complex coordinate vectors, but then $\mathbf a\cdot\mathbf b$ means the bilinear dot product $\sum_i a_i b_i$, not the Hermitian inner product $\sum_i a_i^*b_i$.

Trace Orthogonality

The four matrices $\sigma_\mu$, with $\mu\in\{0,x,y,z\}$, are orthogonal in the Hilbert–Schmidt inner product:

$$\operatorname{tr} (\sigma_\mu^\dagger\sigma_\nu) =2\delta_{\mu\nu}.$$

Useful trace identities include

$$\begin{aligned} \operatorname{tr}(\sigma_i\sigma_j) &=2\delta_{ij},\\ \operatorname{tr}(\sigma_i\sigma_j\sigma_k) &=2i\epsilon_{ijk},\\ \operatorname{tr} (\sigma_i\sigma_j\sigma_k\sigma_\ell) &= 2\bigl( \delta_{ij}\delta_{k\ell} -\delta_{ik}\delta_{j\ell} +\delta_{i\ell}\delta_{jk} \bigr). \end{aligned}$$

The four-factor identity follows by multiplying the first two matrices, multiplying the last two, and using trace orthogonality.

Basis for Two-by-Two Matrices

Every $M\in M_2(\mathbb C)$ has a unique expansion

$$M=m_0I+\mathbf m\cdot\boldsymbol{\sigma},$$

where $m_0,m_x,m_y,m_z\in\mathbb C$. Trace orthogonality extracts the coefficients:

$$\begin{aligned} m_0 &=\frac12\operatorname{tr}M,\\ m_i &=\frac12\operatorname{tr}(\sigma_iM). \end{aligned}$$

For

$$M= \begin{pmatrix} a&b\\ c&d \end{pmatrix},$$

the coordinates are

$$\begin{aligned} m_0&=\frac{a+d}{2}, & m_x&=\frac{b+c}{2},\\ m_y&=\frac{c-b}{2i}, & m_z&=\frac{a-d}{2}. \end{aligned}$$

The matrix $M$ is Hermitian exactly when $m_0$ and all three components of $\mathbf m$ are real. Thus

$$\{I,\sigma_x,\sigma_y,\sigma_z\}$$

is a complex basis for all $2\times2$ matrices and a real basis for the Hermitian ones.

Determinant, Spectrum, and Inverse

Using the Pauli-vector square,

$$\begin{aligned} &(m_0I+\mathbf m\cdot\boldsymbol{\sigma}) (m_0I-\mathbf m\cdot\boldsymbol{\sigma})\\ &\qquad = \bigl(m_0^2-\mathbf m\cdot\mathbf m\bigr)I. \end{aligned}$$

Therefore

$$\det M =m_0^2-\mathbf m\cdot\mathbf m.$$

For a Hermitian matrix, $\mathbf m\in\mathbb R^3$, so the eigenvalues are

$$\lambda_\pm =m_0\pm\lVert\mathbf m\rVert.$$

If $\det M\ne0$,

$$M^{-1} = \frac{ m_0I-\mathbf m\cdot\boldsymbol{\sigma} }{ m_0^2-\mathbf m\cdot\mathbf m }.$$

This is the $2\times2$ adjugate formula written in Pauli coordinates. For a general complex $\mathbf m$, no complex conjugation appears in $\mathbf m\cdot\mathbf m$.

Directional Pauli Matrices and Projectors

Let $\mathbf n\in\mathbb R^3$ be a unit vector and define

$$\sigma_{\mathbf n} =\mathbf n\cdot\boldsymbol{\sigma}.$$

Then

$$\sigma_{\mathbf n}^\dagger =\sigma_{\mathbf n}, \qquad \sigma_{\mathbf n}^2=I.$$

Its eigenvalues are $\pm1$, and its spectral projectors are

$$P_\pm(\mathbf n) =\frac12 \left( I\pm\mathbf n\cdot\boldsymbol{\sigma} \right).$$

They obey

$$\begin{aligned} P_\pm^2&=P_\pm,\\ P_+P_-&=0,\\ P_++P_-&=I,\\ \sigma_{\mathbf n}&=P_+-P_-. \end{aligned}$$

This provides a basis-free way to diagonalize any non-scalar Hermitian $2\times2$ matrix:

$$M =\lambda_+P_+(\widehat{\mathbf m}) +\lambda_-P_-(\widehat{\mathbf m}),$$

where $\widehat{\mathbf m}=\mathbf m/\lVert\mathbf m\rVert$.

Matrix Exponentials

Because $\sigma_{\mathbf n}^2=I$, even and odd powers separate:

$$\begin{aligned} e^{z\sigma_{\mathbf n}} &= \sum_{r=0}^{\infty} \frac{z^{2r}}{(2r)!}I\\ &\quad+ \sum_{r=0}^{\infty} \frac{z^{2r+1}}{(2r+1)!} \sigma_{\mathbf n}\\ &= \cosh z\,I +\sinh z\,\sigma_{\mathbf n}. \end{aligned}$$

With $z=-i\theta/2$,

$$\exp\left( -\frac{i\theta}{2} \mathbf n\cdot\boldsymbol{\sigma} \right) = \cos\frac{\theta}{2}\,I -i\sin\frac{\theta}{2}\, \mathbf n\cdot\boldsymbol{\sigma}.$$

More generally, if

$$M=m_0I+r\sigma_{\mathbf n},$$

then the scalar part commutes with the traceless part and

$$e^M =e^{m_0} \left( \cosh r\,I +\sinh r\,\sigma_{\mathbf n} \right).$$

The limiting case $r=0$ is simply $e^{m_0}I$. See Matrix Functions and Exponentials for the general matrix construction.

Conjugation and Rotations

Define

$$U(\mathbf n,\theta) = \exp\left( -\frac{i\theta}{2} \mathbf n\cdot\boldsymbol{\sigma} \right).$$

This matrix is unitary with determinant one. Conjugation rotates Pauli coordinates:

$$U(\mathbf n,\theta) (\mathbf a\cdot\boldsymbol{\sigma}) U(\mathbf n,\theta)^\dagger = \bigl(R_{\mathbf n}(\theta)\mathbf a\bigr) \cdot\boldsymbol{\sigma},$$

where Rodrigues’ formula is

$$\begin{aligned} R_{\mathbf n}(\theta)\mathbf a &= \mathbf a\cos\theta +(\mathbf n\times\mathbf a)\sin\theta\\ &\quad+ \mathbf n(\mathbf n\cdot\mathbf a) (1-\cos\theta). \end{aligned}$$

The half-angle in $U$ and the full angle in $R_{\mathbf n}$ are both essential. Their group-theoretic meaning is developed in SU(2); the physical spin action belongs to Spin Rotations.

Worked Example

Consider the Hermitian matrix

$$H= \begin{pmatrix} 3&1-i\\ 1+i&1 \end{pmatrix}.$$

Its Pauli coordinates are

$$H=2I+\sigma_x+\sigma_y+\sigma_z.$$

Thus $\mathbf m=(1,1,1)$ and $\lVert\mathbf m\rVert=\sqrt3$. Without solving a characteristic polynomial,

$$\lambda_\pm=2\pm\sqrt3.$$

The spectral projectors are

$$P_\pm =\frac12 \left[ I\pm \frac{ \sigma_x+\sigma_y+\sigma_z }{\sqrt3} \right],$$

so

$$H=(2+\sqrt3)P_+ +(2-\sqrt3)P_-.$$

The determinant is

$$\det H=2^2-3=1,$$

and the inverse follows immediately:

$$\begin{aligned} H^{-1} &=2I-\sigma_x-\sigma_y-\sigma_z\\ &= \begin{pmatrix} 1&-1+i\\ -1-i&3 \end{pmatrix}. \end{aligned}$$

Pauli Strings

For $q$ two-dimensional factors, define a Pauli string

$$\begin{aligned} \Sigma_{\boldsymbol{\mu}} &= \sigma_{\mu_1}\otimes\cdots\otimes\sigma_{\mu_q}, \\ \mu_r &\in\{0,x,y,z\}. \end{aligned}$$

There are $4^q$ such matrices, exactly the complex dimension of $M_{2^q}(\mathbb C)$. Tensor-product trace factorization gives

$$\operatorname{tr} \left( \Sigma_{\boldsymbol{\mu}}^\dagger \Sigma_{\boldsymbol{\nu}} \right) = 2^q\delta_{\boldsymbol{\mu}\boldsymbol{\nu}}.$$

Hence every $2^q\times2^q$ matrix has the expansion

$$M = \sum_{\boldsymbol{\mu}} c_{\boldsymbol{\mu}} \Sigma_{\boldsymbol{\mu}}, \qquad c_{\boldsymbol{\mu}} = \frac{1}{2^q} \operatorname{tr} \left( \Sigma_{\boldsymbol{\mu}}M \right).$$

The formula is useful in spin systems and quantum information, but its matrix ordering depends on the tensor-factor convention. See Tensor Products.

Pauli Group and Stabilizers adds the phase-rich group, signed commuting constraints, stabilized-sector counting, and finite error-signature audits; this page retains the one-qubit matrix calculus and Pauli-string operator basis.

Scope in Quantum Mechanics

The matrices themselves are mathematical objects. Several physical uses add different layers of interpretation:

• spin-$1/2$ components use $S_i=\hbar\sigma_i/2$;
• two-level Hamiltonians use a scalar term plus a Pauli vector;
• one-qubit density operators use a trace-one positive Pauli expansion;
• spin rotations use the SU(2) exponential;
• Pauli strings provide operator bases for many two-level factors.

Those applications should not be conflated. In particular, a Pauli matrix is dimensionless, while a spin component carries units of angular momentum. The canonical two-level dynamics treatment is Pauli-Matrix Hamiltonians.

Numerical Checks

For a computed $2\times2$ matrix, extract Pauli coefficients by traces and reconstruct the matrix. The relative residual

$$\epsilon = \frac{ \lVert M-m_0I-\mathbf m\cdot\boldsymbol{\sigma} \rVert }{ \lVert M\rVert }$$

should be consistent with numerical precision.

For a nominally Hermitian matrix, inspect the imaginary parts of $m_0,m_x,m_y,m_z$. Small imaginary components may be roundoff; large ones signal a genuinely non-Hermitian input or an inconsistent convention. Before comparing Pauli-string coefficients from two codes, verify factor order, basis order, and the sign convention for $\sigma_y$.

Common Mistakes

• Reversing the signs of the off-diagonal entries in $\sigma_y$.
• Forgetting that $\sigma_i\sigma_j$ changes sign when distinct indices are reversed.
• Dropping the factor of two in Pauli commutators.
• Using a Hermitian inner product instead of the bilinear dot product in the complex Pauli-vector multiplication identity.
• Applying Pauli-coordinate eigenvalue formulas to a non-Hermitian matrix while treating the coordinates as real.
• Confusing $\sigma_i$ with the dimensionful spin operator $\hbar\sigma_i/2$.
• Using $\theta$ instead of $\theta/2$ in the SU(2) exponential.
• Assuming the basis ordering of a Pauli string is implicit.
• Applying scalar functions entry by entry rather than through the matrix algebra.

Exercises

1. Verify directly that

   $$\sigma_x\sigma_y=i\sigma_z, \qquad \sigma_y\sigma_x=-i\sigma_z.$$

   Use these products to compute $[\sigma_x,\sigma_y]$ and $\{\sigma_x,\sigma_y\}$.

Solution

Direct multiplication gives

$$\begin{aligned} \sigma_x\sigma_y &= \begin{pmatrix} i&0\\ 0&-i \end{pmatrix} =i\sigma_z,\\[4pt] \sigma_y\sigma_x &= \begin{pmatrix} -i&0\\ 0&i \end{pmatrix} =-i\sigma_z. \end{aligned}$$

Therefore

$$\begin{aligned} [\sigma_x,\sigma_y] &=2i\sigma_z,\\ \{\sigma_x,\sigma_y\} &=0. \end{aligned}$$

2. Decompose

   $$M= \begin{pmatrix} 1&2-i\\ 2+i&3 \end{pmatrix}$$

   in the Pauli basis and find its eigenvalues without expanding a determinant.

Solution

The coefficient formulas give

$$\begin{aligned} m_0&=2, & m_x&=2,\\ m_y&=1, & m_z&=-1. \end{aligned}$$

Thus

$$M=2I+2\sigma_x+\sigma_y-\sigma_z.$$

The Pauli vector has norm

$$\lVert\mathbf m\rVert =\sqrt{2^2+1^2+(-1)^2} =\sqrt6,$$

so

$$\lambda_\pm=2\pm\sqrt6.$$

3. Derive the Pauli-vector identity

   $$\begin{aligned} (\mathbf a\cdot\boldsymbol{\sigma}) (\mathbf b\cdot\boldsymbol{\sigma}) &= (\mathbf a\cdot\mathbf b)I\\ &\quad+ i(\mathbf a\times\mathbf b) \cdot\boldsymbol{\sigma} \end{aligned}$$

   from the component multiplication rule.

Solution

Set $X=(\mathbf a\cdot\boldsymbol{\sigma}) (\mathbf b\cdot\boldsymbol{\sigma})$. Expanding both Pauli vectors gives

$$\begin{aligned} X &= \sum_{i,j}a_i b_j\sigma_i\sigma_j\\ &= \sum_{i,j}a_i b_j\delta_{ij}I\\ &\quad+ i\sum_{i,j,k} a_i b_j\epsilon_{ijk}\sigma_k. \end{aligned}$$

The Kronecker-delta term is

$$\sum_i a_i b_i I =(\mathbf a\cdot\mathbf b)I.$$

The coefficient of $\sigma_k$ in the second term is

$$\sum_{i,j}\epsilon_{ijk}a_i b_j =(\mathbf a\times\mathbf b)_k.$$

Combining the terms proves the identity.

4. Let

   $$U_z(\theta) = \exp\left(-\frac{i\theta}{2}\sigma_z\right).$$

   Show that

   $$U_z(\theta)\sigma_xU_z(\theta)^\dagger = \cos\theta\,\sigma_x +\sin\theta\,\sigma_y.$$

Solution

The exponential formula gives

$$\begin{aligned} U_z(\theta) &=cI-is\sigma_z,\\ c&=\cos\frac{\theta}{2},\\ s&=\sin\frac{\theta}{2}. \end{aligned}$$

Its adjoint is $U_z^\dagger=cI+is\sigma_z$. Therefore

$$\begin{aligned} U_z\sigma_xU_z^\dagger &= (cI-is\sigma_z) \sigma_x (cI+is\sigma_z)\\ &= (c^2-s^2)\sigma_x +2cs\,\sigma_y. \end{aligned}$$

Here

$$\sigma_z\sigma_x=i\sigma_y, \qquad \sigma_x\sigma_z=-i\sigma_y,$$

and $\sigma_z\sigma_x\sigma_z=-\sigma_x$ were used. The double-angle identities $c^2-s^2=\cos\theta$ and $2cs=\sin\theta$ complete the proof.

References

• R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, 2013.
• B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 2nd ed., Springer, 2015.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.