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Sylvester’s matrix equation and Roth’s removal rule

Published online by Cambridge University Press:  01 August 2016

F. Gerrish  and
A. J. B. Ward
F. Gerrish
Affiliation:
43 Roman’s Way, Pyrford, Woking, Surrey GU22 8TR
A. J. B. Ward
Affiliation:
19, Woodside Close, Surbiton, Surrey KT5 9JU
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Extract

In 1952 W. E. Roth published two theorems, one of which has come to be known as Roth’s removal rule and (slightly generalised) goes as follows. [Recall that square matrices M, N are similar when there is an invertible matrix R such that RMR-1 = N. The matrix entries can be elements from any field, although for simplicity we shall call them ‘numbers’.]

Type
Articles
Information
The Mathematical Gazette , Volume 82 , Issue 495 , November 1998 , pp. 423 - 430
Copyright
Copyright © The Mathematical Association 1998

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References

1. Roth, W. E., The equations AX - YB = C and AX - XB = C in matrices, Proc. Amer. Math. Soc. 3 (1952) pp. 392–396.Google Scholar
2. Flanders, H. and Wimmer, H. K., On the matrix equations AX - XB = C and AX - YB = C, SIAM J. Appl. Math. 32 (1977) pp. 707–710.Google Scholar
3. Ward, A. J. B., A novel proof of Roth’s removal rule, Int. J. Math. Educ. Sci. Technol. 24 (1993) pp. 575–578.Google Scholar
4. Sylvester, J. J., Comptes Rendus Acad. Sci. Paris 99 (1884).Google Scholar
5. Nicholson, W. K., Elementary linear algebra with applications (2nd edition) PWS-Kent Publishing Company (1990).Google Scholar
6. Ward, A. J. B. and Gerrish, F., A constructive proof by elementary transformations of Roth's removal theorems in the theory of matrix equations, Int. J. Math. Educ. Sci. Technol. (to appear).Google Scholar
7. Roth, W. E., On direct product matrices, Bull. Amer. Math. Soc. 40 (1934) pp. 461–468.Google Scholar