Undergrad Matrices Commuting with Matrix Exponential

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The discussion revolves around the conditions under which the commutation of matrices A and the matrix exponential e^B implies the commutation of A and B. It is established that while [A, e^B] = 0 if [A, B] = 0, the converse is not necessarily true, as demonstrated through counterexamples involving nilpotent and skew-Hermitian matrices. Participants suggest exploring specific cases, such as diagonal matrices with distinct eigenvalues, to find counterexamples. The conversation also touches on the implications of these relationships in quantum mechanics, particularly regarding conserved quantities and symmetries. Ultimately, the need for additional conditions for the implication [A, e^B] = 0 to lead to [A, B] = 0 remains a central focus of the discussion.
  • #31
Thanks.

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.
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.
 

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