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On Matrix Commutators of Higher Order

Published online by Cambridge University Press:  20 November 2018

D. W. Robinson*
Affiliation:
Brigham Young University
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Let Fn be the collection of n-by-n matrices over a field F. For Y in Fn let ΔY be the mapping on Fn given by XΔY = XYYX. In this paper we study the following

Proposition. Let A and B be in Fn and let m be a positive integer. If BΔxm = 0 whenever XΔAm = 0, then B is a polynomial in A with coefficients in F.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Jacobson, N., Lectures in abstract algebra, vol. II (Princeton, 1953).Google Scholar
2. Lagerstrom, P., A proof of a theorem on commutative matrices, Bull. Amer. Math. Soc, 51 (1945), 535536.Google Scholar
3. Mal'cev, A. I., Foundations of linear algebra (San Francisco, 1963).Google Scholar
4. Marcus, M. and Khan, N. A., On matrix commutators, Can. J. Math., 12 (1960), 269277.Google Scholar
5. Parker, W. V., The matrix equation AX = XB, Duke Math. J., 17 (1950), 4351.Google Scholar
6. Robinson, D. W., A note on k-commutative matrices,]. Mathematical Phys. 2 (1961), 776777.Google Scholar
7. Roth, W. E., On k-commutative matrices, Trans. Amer. Math. Soc, 39 (1936), 483495.Google Scholar
8. Smiley, M. F., Matrix commutators, Can. J. Math., 13 (1961), 353355.Google Scholar
9. Taussky, O., Matrix commutators of higher order, Bull. Amer. Math. Soc, 69 (1963), 738.Google Scholar
10. M. Wedderburn, J. H., Lectures on matrices, Amer. Math. Soc. Colloq. Publ., vol. 17 (1934).Google Scholar
11. Zariski, O. and Samuel, P., Commutative algebra, vol. I (Princeton, 1958).Google Scholar