Pick σx, σy, σz, or identity instantly. Multiply, commute, anticommute, or build Kronecker products easily. See exact complex results with traces and determinants displayed.
| Matrix A | Matrix B | Operation | Expected highlight |
|---|---|---|---|
| σx | σy | Commutator [A,B] | Gives a multiple of σz with i. |
| σz | σz | A · B | Returns identity because σz² = I. |
| σx | Identity I | Anticommutator {A,B} | Returns 2σx since AI + IA = 2A. |
| Custom | σy | Kronecker product A ⊗ B | Produces a 4×4 block matrix output. |
Pauli matrices are the standard 2×2 operators for spin‑1/2 systems. They appear in quantum mechanics, quantum information, and condensed matter models. Because they are Hermitian and traceless (except the identity), they form a compact basis for representing qubit observables and many two‑level Hamiltonians.
The set {σx, σy, σz} plus the identity I spans the space of 2×2 complex matrices. Each σ has eigenvalues +1 and −1, trace 0, and determinant −1. The identity has eigenvalues 1 and 1, trace 2, and determinant 1. These invariants help validate your computations quickly.
Pauli products are highly structured: σiσj = δij I + i εijk σk. When i = j, σi² = I. When i ≠ j, the product is purely off‑diagonal in the basis of Pauli matrices and introduces the imaginary unit through the Levi‑Civita symbol εijk. This calculator shows those complex entries explicitly.
The commutator [σi,σj] = 2i εijk σk encodes the SU(2) Lie algebra. For example, [σx,σy] = 2iσz. The anticommutator {σi,σj} = 2δij I captures orthogonality under the trace inner product. These relations are essential for deriving spin dynamics and simplifying operator expressions.
For any 2×2 matrix A, eigenvalues satisfy λ = (tr(A) ± √(tr(A)² − 4det(A)))/2. The calculator applies this formula for complex matrices, including custom inputs. In quantum settings, trace and determinant are convenient checks for unitary‑like behavior and characteristic polynomial consistency.
The Kronecker product A ⊗ B builds a 4×4 operator acting on a two‑qubit Hilbert space. This is the standard route to construct operators like σz ⊗ I or σx ⊗ σx. The output is presented as a block matrix, matching the tensor product definition used in quantum computing.
Common workflows include validating gate generators, checking spin commutation rules, and confirming algebra in derivations. A quick sanity test is σi² = I and det(σi) = −1. Another is that the commutator of two different Pauli matrices produces the third one, scaled by 2i.
Export tools matter when you are documenting calculations. Use CSV to paste matrices into spreadsheets or lab logs. Use PDF to attach clean tables to assignments and technical notes. Keeping the same decimal setting across runs helps you compare outputs consistently when exploring multiple operator identities.
They represent spin components for a spin‑1/2 particle along x, y, and z axes (up to constants). In qubit language, they describe measurements and rotations around the Bloch‑sphere axes.
σy must be Hermitian and obey the SU(2) algebra with σx and σz. The ±i entries ensure correct commutators and keep eigenvalues real, which is required for observables.
Check invariants: for σx, σy, σz the trace is 0, determinant is −1, and squaring returns the identity. For mixed products, verify [σx,σy] gives 2iσz.
Use it when studying non‑commuting observables, Lie algebra relations, or operator dynamics. Commutators appear in uncertainty relations, Hamiltonian evolution, and spin‑operator simplifications.
It creates a two‑system operator. If A acts on qubit one and B on qubit two, then A ⊗ B acts on the combined space, producing a 4×4 matrix used in two‑qubit models and gates.
Yes. Select “Custom 2×2” and enter real and imaginary parts for each element. The calculator will compute products, commutators, traces, determinants, and eigenvalues using complex arithmetic.
Eigenvalues summarize operator behavior, such as measurement outcomes or stability properties. For 2×2 matrices, they are computed from trace and determinant, which provides a reliable algebraic check on your entries.
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.