The discussion centers on the relationship between matrices A and B, specifically whether the condition ##\left[ A, e^B \right]=0## implies ##[A,B]=0##. Participants explore counterexamples using nilpotent and skew-Hermitian matrices, emphasizing the complexity introduced by the properties of the exponential function. A key insight is that while ##[A,B]=0## guarantees ##\left[ A, e^B \right]=0##, the reverse is not universally true, particularly in cases involving skew-Hermitian matrices. The conversation highlights the need for specific conditions under which the implication holds.
PREREQUISITESMathematicians, physicists, and students of linear algebra interested in the properties of matrix commutation, particularly in the context of quantum mechanics and advanced matrix theory.
And if you want to kill the last "by-foot" step, you could substitute 4) by the adjoint representations and that exp is a local diffeo around 0.StoneTemplePython said:Looking at this in terms of a power series really misses the mark as we have powerful diagonalization theorems that we may lean on.