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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

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