On Almost Commuting Matrices

When mathematicians encounter a binary operation, one of the first things they ask is "when does the operation commute?" That is, given an operation * when does one have A*B=B*A? Some operations always commute. Addition and multiplication in the real numbers are examples of this. Sometimes they commute under certain restricted circumstances. For example, subtraction rarely commutes (1-2 is not the same as 2-1).

In high school, children are often taught about matrices and matrix multiplication. Matrix multiplication seems to be given in part simply to have an example of an operation which has complicated behavior regarding when it commutes. Of course, we can only meaningfully talk about this when the matrices in question are assumed to be square matrices since otherwise the multiplication won't even be defined. However, matrices have other operations which we can do on them other than just matrix addition and multiplication. In particular, we can also multiply a matrix by a scalar.

This raises the following question: When do matrices almost commute? By almost commute, we mean commute up to multiplication by a non-zero scalar. A general result seems difficult. But there is at least one pretty result which can be proven without too much trouble by looking at the eigenvalues and eigenvectors of a pair of matrices:

If cAB=BA for some constant c, and A is invertible, then c is a dth root of unity for some d such that d divides the number of distinct non-zero eigenvalues of B. It isn't that hard to generalize this result slightly with A not invertible. However, one then needs the slightly technical condition that A sends no non-zero eigenvector of B to zero. Note also that this result is most nicely stated in the slightly more restricted symmetric case when both A and B are invertible.

One pretty corollary of this result is that if A and B are invertible p x p matrices over the real numbers where p is an odd prime, with all distinct eigenvalues, then A and B are almost commuting if and only if they actually commute.