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.
Topics
Complex analysis
REFERENCES
1.
For a survey of these results see
O.
Taussky
, Am. Math. Monthly
64
, 229
(1957
).2.
3.
See also,
M.
Marcus
and N. A.
Khan
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64B
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(1960
),4.
For the definition and properties of the principal idempotents of a matrix, see, for example, N. Jacobson, Lectures in Abstract Algebra (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1953), Vol. II, pp. 130–132.
5.
6.
Reference 5, Theorem 5.
7.
Reference 5, Theorem 3.
8.
For a proof of this corollary, in case either or see also
T.
Kato
and O.
Taussky
, J. Wash. Acad. Sci.
46
, 38
(1956
).9.
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© 1961 American Institute of Physics.
1961
American Institute of Physics
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