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The Matrix Equations A = XYZ And B = ZYX and Related Ones

Published online by Cambridge University Press:  20 November 2018

J. L. Brenner  and
M. J. S. Lim
J. L. Brenner
Affiliation:
Department of Mathematics, University of Victoria, Victoria, B.C., Canada
M. J. S. Lim
Affiliation:
J. L. Brenner 10 Phillips Rd. Palo Alto, California94303
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In [15], O. Taussky-Todd posed the problem of title, namely to find X, Y, Z when A, B are given. Clearly if X, Y, Z exist then A, B are either both invertible or both noninvertible.

In section 1, the problem is reviewed in case A, B are both invertible. The problem is seen to be fundamentally one of group theory rather than matrix theory. Application of results of Shoda, Thompson, Ree to the general group-theoretical results allows specialization to certain matrix groups.

In Section 2, examples and counterexamples are given in case A, B are noninvertible. A general necessary condition for solvability (involving ranks) is obtained. This condition may or may not be sufficient. For dim A=2, 3 the problem is settled: there is always a solution in the noninvertible case.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 2 , 01 June 1974 , pp. 179 - 183
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Brenner, J. L., Math. Reviews 16, 326 (1955).Google Scholar
1a. Brenner, J. L., Some Simultaneous equations in matrices. Duke Math. J. 39, 801-805 (1972).Google Scholar
2. Brown, A. and Pearcy, C., Operators of the Form PAQ-QAP, Can. J. Math. 20 (1968), 1353- 1361.Google Scholar
3. de Pillis, J. and Brenner, J. L., Generalized elementary symmetric functions and quaternion matrices, Linear Algebra and its Applications, 4, 55-69 (1971).Google Scholar
4. Flanders, H., Elementary Divisors of AB and BA, Proc. Amer. Math. Soc. 2, 871-874 (1951).Google Scholar
5. Griffiths, H. B., A note on commutators in free products, Proc. Camb. Philos. Soc. 50, 178-188 (1954).Google Scholar
6. Hall, P.. Correspondence.Google Scholar
6a. Lyndon, R. C., The equation a2b2 = c2 in free groups. Mich. Math. J. 6, 89-95 (1959).Google Scholar
7. Neumann, B.. Correspondence.Google Scholar
8. Ree, R., Commutators in semisimple algebraic groups, Proc. Amer. Math. Soc. 15, 457-460 (1964).Google Scholar
9. Shoda, K., Einige Sätze uber Matrizen, Jap. J. Math. 13, 361-365 (1936).Google Scholar
10. Shoda, K., Über den Kommutator der Matrizen, J. Math. Soc. Japan 3, 78-81 (1951).Google Scholar
11. Thompson, R. C., Commutators in the special and general linear groups, Trans. Amer. Math. Soc. 101, 16-33 (1961).Google Scholar
12. Thompson, R. C., Commutators of matrices with coefficients from the field of two elements, Duke J. 29, 367-374 (1962).Google Scholar
13. Thompson, R. C.. On matrix commutators, Portugaliae Math. 21, 143-153 (1962).Google Scholar
14. Taussky-Todd, O., Generalized commutators of matrices and permutations of factors in a product of three matrices, Studies in Math. Mech. presented to R. von Mises, N.Y. Academic Press, 1954, 67-68.Google Scholar
15. Taussky-Todd, O., Problems on matrices and operators, Bull. Amer. Math. Soc. 76, 977 (1970).Google Scholar
16. Wicks, M. J., Commutators in free products, J. London Math. Soc. 37, 433-444 (1962).Google Scholar