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Dedicated to the memory of
Olga Taussky-Todd
Let A and B be n × n matrices, let C = ABA−1B−1 be the multiplicative
commutator of A and B, and assume AC = CA. Olga Taussky (1961) examined the
structure of A and B for the case where A and B are unitary. Marcus and Thompson
(1966) generalized her results to the case where A and C are assumed normal. We
look at this problem for general A, with particular attention to the cases where A is
diagonalizable or nonderogatory.
Now let [A,B] = AB −BA be the additive commutator of A and B and assume
A commutes with [A,B]. The finite-dimensional version of a theorem of Putnam tells
us that if A is normal, then A and B commute. We show that the same conclusion
holds when A is diagonalizable. If A is nonderogatory, then A and B can be
simultaneously triangularized.
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