Matrices that *almost* commute

Abstract:

In this talk we will look at square matrices A and B that satisfy that AB is close to BA. That is to say, matrices A and B that almost commute. I will explain what it means (to operator algebraists) that two matrices are close. We will look at the questions of whether one can find exactly commuting matrices close to almost commuting ones. We also look at how this question changes if we impose additional conditions on our matrices, say that they are unitary or real or something else.

The question has drawn interest from some mathematicians just because it is a natural question. Lately questions of this form has also drawn interest from physicists due to its application in the field of so-called topological insulators. I will attempt to justify the connection to physics. I will only assume basic knowledge a of linear algebra and little familiarity with continuity.