On approximately simultaneously diagonalizable matrices

Some thoughts on Gerstenhaber's theorem

“…Suppose A, B, C can be perturbed to simultaneously diagonalizable matrices A, B, C (such as A having n distinct eigenvalues and B, C commuting with A). Then dim C[A, B, C] ≤ n. But for sufficiently small perturbations, dim C[A, B, C] ≤ dim C[A, B, C] and therefore dim C[A, B, C] ≤ n. See Section 2 of [15] or Theorem 6.3.3 of [16].…”

On a conjecture of Tomas Sauer regarding nested ideal interpolation

“…As the three operators in (3.3) commute with the 2-regular operator µ x + µ z , they can be approximated by commuting simultaneously diagonalizable operators (cf. [7,Theorem 6.1]). This, in turn, implies (cf.…”

The tropical commuting variety

“…Algebro-geometrically, one can study the commuting variety, which is generated by the n 2 equations (XY ) ij − (Y X) ij = 0. These perspectives and many other variants have been studied in the classical setting [OCV,§5]. This paper considers similar questions for tropical and tropicalized matrices.…”

“…Commutativity of tropical matrices is one such example. Classically, if A, B ∈ C n×n where A has n distinct eigenvalues, then AB = BA if and only if B can be written as a polynomial in A [OCV,§5]. Moreover, if B has n distinct eigenvalues, then AB = BA if and only if A and B are simultaneously diagonalizable.…”