On the number of invariant factors of matrix products (Q1779407)

Let \(\mathbb{F}\) denote an arbitrary field and \(\overline{\mathbb{F}},\) its algebraic closure. Let \(A\) be an \(n\times n\) matrix over \(\mathbb{F}\) and let \({\mathcal A}\) represent the set of all matrices over \(\mathbb{F}\) similar to \(A,\) that is, \({\mathcal A}\) represents the similarity class of \(A\). Let \(i^*({\mathcal A})\) denote the number of invariant factors of \({\mathcal A}\) with at least one nonzero root in \(\overline{\mathbb{F}}.\) The authors characterize the classes \({\mathcal A}\) and \({\mathcal B},\) for which there exists \(A\in {\mathcal A}\) and \(B\in {\mathcal B}\) such that \(i^*(AB)=0\) when all eigenvalues of \({\mathcal A}\) and \({\mathcal B}\) lie in \(\mathbb{F}.\) The authors also characterize the classes \({\mathcal A}\) and \({\mathcal B},\) for which there exist \(A\in {\mathcal A}\) and \(B\in {\mathcal B}\) such that \(i^*(AB)=i^*({\mathcal A})+i^*({\mathcal B})-n\) over algebraically closed fields.