A Note on k-Commutative Matrices

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Abstract
Let A and B be square matrices over a field in which the minimum polynomial of A is completely reducible. It is shown that A is k commutative with respect to B for some non‐negative integer k if and only if B commutes with every principal idempotent of A. The proof is brief, simplifying much of the previous study of k‐commutative matrices. The result is also used to generalize some well‐known theorems on finite matrix commutators that involve a complex matrix and its transposed complex conjugate.
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