Analysis seminar

George Hutchinson, Department of Pure Math, University of Waterloo

“Almost Commuting Matrices and Lin’s Theorem”

Suppose that two finite rank, self-adjoint matrices A and B have small commutator—that is, the norm of AB−BA is small. Are there necessarily two finite rank, self-adjoint matrices A′ and B′ that commute with each other such that A′ is sufficiently close to A and B′ is sufficiently close to B? We will prove the titular theorem (which answers this in the affirmative) as well as examining a similar problem regarding unitary matrices. If time permits, we will also hint at the power of Lin’s Theorem by using it to give an easy proof of the Weyl-von Neumann-Berg Theorem.