Geodesic
The Matrix Equations A = XYZ And B = ZYX and Related Ones
Canadian mathematical bulletin, Tome 17 (1974) no. 2, pp. 179-183

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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. 
Brenner, J. L.; Lim, M. J. S. The Matrix Equations A = XYZ And B = ZYX and Related Ones. Canadian mathematical bulletin, Tome 17 (1974) no. 2, pp. 179-183. doi: 10.4153/CMB-1974-036-6
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