A note on matrix solutions to $A=XY-YX$
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- by Charles R. Johnson
- Proc. Amer. Math. Soc. 42 (1974), 351-353
- DOI: https://doi.org/10.1090/S0002-9939-1974-0332826-1
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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.References
- A. A. Albert and Benjamin Muckenhoupt, On matrices of trace zeros, Michigan Math. J. 4 (1957), 1β3. MR 83961
- Shmuel Friedland, Matrices with prescribed off-diagonal elements, Israel J. Math. 11 (1972), 184β189. MR 379526, DOI 10.1007/BF02762620
- Fergus Gaines, A note on matrices with zero trace, Amer. Math. Monthly 73 (1966), 630β631. MR 199207, DOI 10.2307/2314800
- W. W. Parker, Sets of complex numbers associated with a matrix, Duke Math. J. 15 (1948), 711β715. MR 26985 K. Shoda, Einige SΓ€tze ΓΌber Matrizen, Japan J. Math. 13 (1936), 361-365.
Bibliographic Information
- © Copyright 1974 American Mathematical Society
- 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