On approximately simultaneously diagonalizable matrices

A collection A1, A2, …, Ak of n × n matrices over the complex numbers has the ASD property if the matrices can be perturbed by an arbitrarily small amount so that they become simultaneously diagonalizable. Such a collection must perforce be commuting. We show by a direct matrix proof that the ASD property holds for three commuting matrices when one of them is 2-regular (dimension of eigenspaces is at most 2). Corollaries include results of Gerstenhaber and Neubauer–Sethuraman on bounds for the dimension of the algebra generated by A1, A2, …, Ak. Even when the ASD property fails, our techniques can produce a good bound on the dimension of this subalgebra. For example, we establish for commuting matrices A1, …, Ak when one of them is 2-regular. This bound is sharp. One offshoot of our work is the introduction of a new canonical form, the H-form, for matrices over an algebraically closed field. The H-form of a matrix is a sparse “Jordan like” upper triangular matrix which allows us to assume that any commuting matrices are also upper triangular. (The Jordan form itself does not accommodate this.)