The Matrix EquationsA = XYZAndB = ZYXand Related Ones

Abstract

In [15], O. Taussky-Todd posed the problem of title, namely to findX, Y, ZwhenA, Bare given. Clearly ifX, Y, Zexist thenA, Bare either both invertible or both noninvertible.In section 1, the problem is reviewed in caseA, Bare 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 caseA, Bare noninvertible. A general necessary condition for solvability (involving ranks) is obtained. This condition may or may not be sufficient. For dimA=2, 3 the problem is settled: there is always a solution in the noninvertible case.