# Applications of measurement of commutativity

• I
In summary, the commutator is an operation on two linear operators that measures the gap between flowing along the two operators in different orders. It is often used as a heuristic for measuring the commutativity of matrices, but it needs to be defined on a curved space to be useful. This has a fundamental relationship to trajectories and derivatives on manifolds, as explained in the link provided.

Dang this place has a topology and analysis section too, nice.

This is probably a graduate level topic, but I am by no means an expert on these subjects, just things I learn from wikipedia and other people. The commutator is an operation on two linear operators (most often matrices) of the form ##[A,B] = AB - BA.## This is often touted as a measurement of how "badly" two matrices fail to commute, but I don't think it quite is.

I'd like to research what happens when we induce a matrix norm to the commutator, when the matrix norm of the commutator ##|| [A,B] ||^2## actually returns a specific number, I think that's more of a "measurement" of the commutativity of two matrices.

However, why bother? Are there any theoretical or scientific applications for measuring the size of matrix commutators? Possibly in quantum physics, though I don't know that subject in depth, I hope there are more applications than that.

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You have to consider ##A## and ##B## as vector fields. ##[A,B]## measures the gap you get if you flow along ##A## and then ##B##, or the other way around. It doesn't measure how badly matrices commute, it measures how commutative flows along vector fields are.

• topsquark
Okay, that's interesting. What about measuring the commutativity of matrices instead of vectors though? I haven't seen the commutator in the case that ##A## and ##B## are vector fields, but rather matrices. • 