Flandersʼ theorem for many matrices under commutativity assumptions

Abstract

We analyze the relationship between the Jordan canonical form of products, in different orders, of k square matrices A1,…,Ak. Our results extend some classical results by H. Flanders. Motivated by a generalization of Fiedler matrices, we study permuted products of A1,…,Ak under the assumption that the graph of non-commutativity relations of A1,…,Ak 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 difference 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.