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A reduction for the matrix equation AB = εBA

Published online by Cambridge University Press:  24 October 2008

M. P. Drazin
M. P. Drazin
Affiliation:
Trinity CollegeCambridge
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Extract

1. It is well known that, if two n × n matrices A, B commute, then there is a non-singular matrix P such that P−1AP, P−2BP are both triangular (i.e. have all their subdiagonal elements zero). This result has been generalized, and, in particular, has been shown to hold even if the commutator K = AB − BA is not zero, provided that K is properly nilpotent in the polynomial ring generated by A, B.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 47 , Issue 1 , January 1951 , pp. 7 - 10
Copyright
Copyright © Cambridge Philosophical Society 1951

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References

* Cayley, A., ‘A memoir on the theory of matrices’, Philos. Trans. 148 (1858), 17–37CrossRefGoogle Scholar; Coll. Works (Cambridge, 1889), 2, 475–96.Google Scholar

† The general case has also been considered, from a different point of view, by Cecioni, F., ‘Sull’ equazione fra matrici AX = εXA’, Ann. Univ. Toscane, 14 (1931), fasc. 2, 1–49Google Scholar; Cherubino, S., ‘Sulle omagrafie permutabili’, Rend. Semin. mat. Roma (4), 2 (1938), 14–46Google Scholar; and Kurosaki, T., ‘Über die mit einer Kollineation vertauschbaren Kollineationen’, Proc. Imp. Acad. Tokyo, 17 (1941), 24–8.CrossRefGoogle Scholar

* The condition of Lemma 2 is, in fact, also sufficient; cf. Wilson, R., ‘The equation px = xq in linear associative algebra’, Proc. London Math. Soc. (2), 30 (1930), 359–66.CrossRefGoogle Scholar