Generalization of Flanders' theorem to matrix triples. (Q1428618)

This paper deals with several ways to generalize the Flanders' theorem to matrix triples. The authors consider six invertible matrices and try to write them as the possible products of three matrices, that is, given \(B_1, B_2, \ldots, B_6 \in M_n(C)\) invertible matrices they consider the matrix equations \[ \begin{matrix} B_1=A_1A_2A_3,& &B_2=A_2A_3A_1,& & B_3=A_3A_1A_2, \\ B_4=A_1A_3A_2,& &B_5=A_2A_1A_3,& & B_6=A_3A_2A_1, \end{matrix} \] where \(A_1, A_2, A_3 \in M_n(C)\) are unknown matrices. Initially, the authors describe a wide set of necessary conditions so that this system be solvable, showing that they are not sufficient. Next, they study the simultaneous solvability of two equations, selected appropriately among the matrix system. The rest of the paper is devoted to the study of a particular case, in which the six given matrices are simultaneously diagonalizable, with distinct nonzero eigenvalues. In this case, the authors obtain a necessary and sufficient condition for the solvability of the full matrix system. Moreover, they construct an explicit solution of the mentioned system.