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It is known that a nonsingular, nonscalar, n-by-n complex matrix A may be factored as A = BC, in which the spectra of B and C are arbitrary, subject to det(A) = det(B)det(C). Furthermore, it is also known that B and C can be taken to be nonderogatory. Additionally it has been shown that when two matrices have eigenvalues of high geometric multiplicity, this restricts the possible Jordan structure of the third. We demonstrate a previously unknown restriction on the Jordan structures of B and C.doi:10.1142/s1793557112500180 fatcat:uccsegqyl5butal6fce26mo5gy