Flanders' theorem for many matrices under commutativity assumptions

We analyze the relationship between the Jordan canonical form of products, in different orders, of $k$ square matrices $A_1,...,A_k$. Our results extend some classical results by H. Flanders. Motivated by a generalization of Fiedler matrices, we study permuted products of $A_1,...,A_k$ under the assumption that the graph of non-commutativity relations of $A_1,...,A_k$ is a forest. Under this condition, we show that the Jordan structure of all nonzero eigenvalues is the same for all permuted products. For the eigenvalue zero, we obtain an upper bound on the dierence between the sizes of Jordan blocks for any two permuted products, and we show that this bound is attainable. For $k = 3$ we show that, moreover, the bound is exhaustive