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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A note on matrix solutions to $ A=XY-YX$


Author: Charles R. Johnson
Journal: Proc. Amer. Math. Soc. 42 (1974), 351-353
MSC: Primary 15A24
DOI: https://doi.org/10.1090/S0002-9939-1974-0332826-1
MathSciNet review: 0332826
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Abstract: It is known that a square matrix A can be written as a commutator $ XY - YX$ if and only if $ {\operatorname{Tr}}(A) = 0$. In this note it is shown further that for a fixed A the spectrum of one of the factors may be taken to be arbitrary while the spectrum of the other factor is arbitrary as long as the characteristic roots are distinct. The distinctness restriction on one of the factors may not in general be relaxed.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0332826-1
Keywords: Commutator, trace, eigenvalues, positive definite
Article copyright: © Copyright 1974 American Mathematical Society