Physical Sciences and Mathematics Commons^™

A Note On K-Commutative Matrices, D. W. Robinson

Faculty Publications

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.