We know that for matrices A and B, AB is not the same as BA in general. But suppose B is a polynomial in A, like B = A^2 - 3A + I (note I = A^0). Then AB is indeed the same as BA. Some natural questions follow for a set S of (square) matrices that commute with each other.
1. Does there always exist an A in S that "generates" the rest via polynomials?
2. Since diagonal matrices commute with each other, is it the case that we can (simultaneously) diagonalise all matrices in S?
3. If (1) and (2) are false, then what do these non-trivial sets S look like?
In this talk we will see an elegant characterisation of commuting matrices which follows from the works of Marinari, Möller and Mora (1993), and Möller and Stetter (1995). Similar in spirit to question (1) above, the characterisation involves multivariate polynomials and uses a notion of "multiplicity of a polynomial at a point" that is slightly non-standard in CS.