Paste generators, choose dimensions, and compute brackets. Inspect products, traces, norms, commuting status, and magnitudes. Visualize structure, export results, and study worked physics examples.
Enter square matrices with comma-separated entries and one row per line. Use real or complex values such as 2, -3.5, i, or 4-2i.
Use the responsive field grid below. Large screens show three columns, smaller screens show two, and phones show one.
| Example | Matrix A | Matrix B | Expected commutator | Physics note |
|---|---|---|---|---|
| Pauli pair | σx = [[0,1],[1,0]] | σy = [[0,-i],[i,0]] | 2iσz | Shows noncommuting spin generators. |
| Diagonal pair | diag(2,3) | diag(5,-1) | Zero matrix | Any diagonal pair commutes. |
| Planar rotation basis | [[0,-1],[1,0]] | [[1,0],[0,-1]] | [[0,2],[2,0]] | Useful for symmetry practice. |
Lie algebra commutator: [A,B] = AB - BA
Frobenius norm: ||[A,B]||F = √Σ|cij|²
Trace diagnostic: tr([A,B]) = 0 for finite-dimensional matrices.
Jacobi identity check: [A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0
These relations help verify whether your operator matrices behave like a consistent Lie algebra representation.
a+bi, a-bi, i, or -i.A zero commutator means the two operators commute, so applying them in either order produces the same matrix action. In physics, that often signals compatible observables or a shared eigenbasis.
Quantum generators, spin matrices, and many symmetry operators naturally contain imaginary components. Complex support lets the tool handle Pauli matrices, ladder operators, and other realistic physics examples without manual conversions.
For finite-dimensional matrices, the trace of a commutator should vanish. This gives a quick numerical sanity check. Tiny nonzero values usually come from rounding or finite precision, not a failed algebraic identity.
The Frobenius norm measures the overall size of the commutator. It is useful when entries nearly cancel, because it compresses the full matrix into one scalar that quantifies how strongly noncommuting the operators are.
The extra matrix allows a Jacobi identity check. That identity is fundamental for Lie algebras, so evaluating the Jacobi sum helps confirm whether your chosen matrices behave like a consistent bracket representation.
Yes. The calculator is suitable for angular momentum operators, Pauli matrices, small representation matrices, and classroom symmetry exercises where commutators reveal structure constants or conservation relationships.
Type one row per line and separate entries with commas. Good examples include 1, 0 or 2+i, -3. Make sure the number of rows and columns matches the chosen dimension exactly.
A very small norm means the matrices commute within your selected tolerance. Numerical rounding, decimal truncation, or approximate inputs can create tiny residuals, so tolerance helps separate genuine noncommutation from harmless computation noise.
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.