Compute matrix commutators for generators and operators. Test noncommuting symmetry actions in quantum models quickly. Export neat tables, verify Jacobi identity, and share results.
For two generators or operators represented as matrices A and B, the commutator is:
[A,B] = AB − BA
If [A,B] = 0, the pair commutes. Nonzero values encode structure and symmetry in many physical models, including quantum dynamics and gauge theory.
Example with 2×2 matrices (Pauli-like generators). Try it using the Fill example buttons.
| Matrix | Row 1 | Row 2 | ||
|---|---|---|---|---|
| A | 0 | 1 | 1 | 0 |
| B | 0 | -1 | 1 | 0 |
| [A,B] | 2 | 0 | 0 | -2 |
Lie brackets appear whenever continuous symmetries are studied through infinitesimal transformations. In practice you often work with a concrete matrix representation, such as spin operators, angular momentum, ladder operators, or gauge generators. This page helps you compute commutators, anticommutators, and a Jacobi check in a consistent numeric workflow. Use it to compare conventions, verify closure of a proposed generator set, and build intuition about how algebraic relations map to physical behavior.
In many physical theories, symmetries are encoded by generators that do not commute. The commutator captures how two infinitesimal transformations fail to cancel, and this failure is the algebraic fingerprint of the symmetry group.
Quantum observables and symmetry generators can be represented as matrices once a basis is chosen. The bracket [A,B] then becomes ordinary matrix multiplication and subtraction, letting you explore algebraic structure numerically and consistently.
A zero commutator indicates compatible measurements or commuting symmetry actions. A nonzero commutator signals coupling between directions in the algebra. Diagonal patterns often indicate a generator that behaves like a Cartan element, while off‑diagonal entries reveal mixing.
Different conventions use different overall factors for generators. The scalar option helps match common normalizations, such as factors of i in quantum mechanics or rescalings used to compare structure constants between references and models.
Any Lie algebra must satisfy [A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0. With numeric matrices, the Jacobi combination should be near zero up to rounding. When assessing results, compare max |J| to typical entry sizes in A, B, and C and choose a tolerance (for example 1e-10 to 1e-6) appropriate to your scale. Large deviations usually point to inconsistent inputs or numerical error.
With Pauli-like 2×2 inputs, you can reproduce classic relations used in spin physics. For example, choosing two different Pauli matrices yields a commutator proportional to the third, illustrating closure of the algebra and why commutators generate rotations.
Floating‑point arithmetic can produce small nonzero values even when theory predicts exact zeros. Use the trace and the maximum absolute Jacobi entry as quick diagnostics. If values are tiny compared to input magnitudes, they are typically numerical noise.
This calculator is useful for testing candidate generator sets, checking closure under commutation, and validating matrix representations used in simulations. Exporting tables makes it easy to document algebra checks in reports, lab notes, or course material.
If [A,B]=0, the two matrices commute. In physics this often means the corresponding operations are compatible, or the observables can share a common eigenbasis in that representation.
The bracket measures noncommutativity. Subtracting the reverse product isolates the antisymmetric part of multiplication, which is the defining property needed for Lie algebra structure.
The anticommutator {A,B}=AB+BA appears in fermionic algebra, Clifford relations, and uncertainty calculations. It complements the commutator by capturing the symmetric part of operator products.
Use it to match your convention. Some texts absorb factors of i into generators, while others keep them explicit. Scaling helps compare results across conventions without changing inputs.
It tests whether three inputs behave like Lie algebra elements under commutation. For exact matrix representations, the Jacobi result should be zero; small residuals typically come from rounding.
Floating‑point rounding and subtraction of close numbers can create small artifacts. Reduce input magnitude, avoid unnecessary scaling, and treat values near machine precision as numerical noise.
This tool evaluates numeric matrices only. For symbolic brackets, use a computer algebra system, then substitute numeric matrices here to verify a specific representation or test numeric closure.
Note: This tool uses numeric matrices. For symbolic Lie brackets, use numeric substitutions or a CAS workflow.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.