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On Matrix Commutators

Published online by Cambridge University Press:  20 November 2018

Marvin Marcus  and
Nisar A. Khan
Marvin Marcus
Affiliation:
The University of British Columbia and Muslim University, Aligarh
Nisar A. Khan
Affiliation:
The University of British Columbia and Muslim University, Aligarh
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Let A, B, and X be n-square matrices over an algebraically closed field F of characteristic 0. Let [A, B] = AB — BA and set (A, B) = [A, [A, B]]. Recently several proofs (1; 3; 5) of the following result have appeared: if det (AB) ≠ 0 and (A,B) = 0 then A-1B-1AB - I is nilpotent. In (2) McCoy determined the general form of any X satisfying

1.1

in the case that A has a single elementary divisor corresponding to each eigenvalue, that is, A is non-derogatory. In Theorem 1 we determine the structure of any matrix X satisfying (1.1) and also give a formula for the dimension of the linear space of all such X in terms of the degrees of the elementary divisors of A.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 269 - 277
Copyright
Copyright © Canadian Mathematical Society 1960

References

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