Understanding the Commutator of Matrices

The commutator of two matrices A and B, denoted by [A,B], is a matrix that represents the result of applying the operation of one matrix to the other. Specifically, [A,B] = AB -BA. In other words, it is the difference between the product of A and B, and the product of B and A.

For example, if we have two matrices A = [a11, a12; a21, a22] and B = [b11, b12; b21, b22], then the commutator [A,B] = AB -BA would be:

[A,B] = [a11b11 + a12b21, a11b12 + a12b22; a21b11 + a22b21, a21b12 + a22b22] - [b11a11 + b12a21, b11a12 + b12a22; b21a11 + b22a21, b21a12 + b22a22]

= [a11b22 - b11a22, a12b21 - b12a21; a21b12 - b21a12, a22b11 - b22a11]

The commutator of two matrices can be used to measure the failure of the matrix product to commute. If the commutator is zero, then the matrix product commutes, meaning that the order in which we multiply the matrices does not matter. If the commutator is non-zero, then the matrix product does not commute, and the order in which we multiply the matrices does matter.

In summary, the commutator of two matrices is a measure of how well the matrix product commutes, and can be used to determine if the product is commutative or not.