Matrix Commutator

Edit two real 2x2 matrices to compare AB and BA, visualize their action on one vector and distinguish a zero commutator from its always-zero trace.

Matrix Commutator: measuring the difference in composition order

Matrix multiplication is generally not commutative. Applying transformation B then A can yield a different result than applying A then B. The matrix commutator C = AB − BA measures this difference. If C is the zero matrix, the matrices commute. This experiment lets you compare the action of AB and BA on a vector, and observe the resulting difference vector.

Worked example: matrix units

Consider the "Matrix units" preset: A = [[0, 1], [0, 0]] and B = [[0, 0], [1, 0]]. Let us compute the commutator and apply it to the vector v = (1, 1).

First, calculate AB: the product is [[1, 0], [0, 0]]. Applying this to v = (1, 1) yields ABv = (1, 0).
Next, calculate BA: the product is [[0, 0], [0, 1]]. Applying this to v = (1, 1) yields BAv = (0, 1).
Now compute the commutator C = AB − BA = [[1, 0], [0, -1]].
The trace of C is the sum of its diagonal elements: 1 + (-1) = 0.
Applying C to v = (1, 1) yields Cv = (1, -1). This matches the difference vector ABv − BAv = (1, 0) − (0, 1) = (1, -1).

Experimenting with the commutator

Open Matrix Commutator. You can select presets or edit individual matrix and vector entries to observe their effects.

Select the "Diagonal and swap" preset. The commutator matrix C becomes [[0, -1], [1, 0]]. With v = (1, 1), you can see Cv = (-1, 1).
Click "Swap A and B". Observe that the entries of C are all negated, becoming [[0, 1], [-1, 0]], and the vector Cv reverses direction.
Change the preset to "Identity". Because the identity matrix commutes with any matrix, C becomes the zero matrix, and trace(C) is 0. Both ABv and BAv coincide on the canvas.
Edit the vector to (0, 0). All transformation vectors collapse to the origin, regardless of whether the matrices commute.

Properties and edge cases

A key property of the matrix commutator is that its trace is always zero, regardless of the matrices A and B. This is because trace(AB) = trace(BA) for any square matrices. However, as demonstrated by the "Matrix units" example, a zero trace does not imply that the matrices commute (i.e., that C is the zero matrix). The trace is zero, but C itself is nonzero.

Other boundary cases include equal matrices (A = B), where the commutator is trivially zero. Furthermore, a zero vector (0, 0) will result in coincident output vectors even if the matrices do not commute. Therefore, visual coincidence of ABv and BAv for a single vector does not guarantee that the matrices commute; one must inspect the commutator matrix C itself.

The canvas adapts a common scale to keep all four vectors in view, preserving their relative lengths. Drag to pan and use Reset view to recenter; this does not change the matrices. Displayed matrix/vector components are rounded to two decimals, and the commute indicator uses a small floating-point tolerance. The interface handles blank, nonfinite, or out-of-range inputs with an explicit validation state, preventing mathematically undefined behaviors without coercing inputs to zero.

References

For a formal definition and properties of the matrix commutator, see the NIST Digital Library of Mathematical Functions: DLMF 1.2(vi) Matrices: Commutator.